The group of neat extensions

THE GROUP OF NEAT EXTENSIONS IliilU -J o -4 \J\

^ 2 :

THE GROUP

OF NEAT EXTENSIONS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H. R. VAN NAUTA LEMKE, HOOGLERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP

DONDERDAG 15 OKTOBER 1970 TE 16 UUR

DOOR

MARIUS JOHANNES SCHOEMAN MASTER OF SCIENCE

GEBOREN TE PRETORIA

I CjU

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. F. LOONSTRA.

Hiermee betuig ek my dank aan die

SUID-AFRIKAANSE WETENSKAPLIKE EN NYWERHEIDNAVORSINGSRAAD en die

UNIVERSITErr VAN PRETORIA

C O N T E N T S

Table of notations viii

CHAPTER I Introduction 1

1 Preliminaries 2 2 The extension problem 5

CHAPTER II Neat subgroups and extensions 9

3 Neat subgroups 9 4 Neat extensions 14

CHAPTER III Homological methods 20 5 Neat extensions as short exact sequences 20

6 Some properties of Next 30 7 Groups that are groups of neat extensions 34

CHAPTER IV The Frattini subgroups of Ext(5, A) 38

8 «-Neat subgroups and extensions 38 9 The «-th Frattini subgroup of Ext (5, ^ ) 39

CHAPTER V Splitting extensions by torsion groups 41 10 Groups all of whose extensions by torsion groups are

splitting 41 11 P,-groups 42 12 Ext (5, ^ ) and algebraic compactness 44

References 46 Samenvatting 47 Biography 48

TABLE OF NOTATIONS

P The set of all prime numbers N The set of all natural numbers

Ö The field of rational numbers Z The ring of integers

Ax B The cartesian product of a set A and a set B A®B The direct sum of the groups A and B @ Ai The direct sum of the groups Ai, iel iel

Y\ Ai The (unrestricted) direct product of the groups Ai, iel

I E /

Z (p*") The quasicyclic group belonging to the prime p

Z{n) The cyclic group of finite order «

{x} The cyclic group generated by the element x {A, B} The group generated by the groups A and B

Ker(>7) The kernel of the homomorphism rj 1^ The identity automorphism of the group A

A [n] The subgroup of the group A consisting of those elements a

for which na = 0,neN

tA The maximal torsion subgroup of the group A ^{A) The Frattini subgroup of the group A

ERRATA p. 18 lines 26 and 29: peP read: peP p. 19 line 5, read:

Furthermore, since G/pG is elementary we have G/pG=®Z{j>). p. 36 lines 25, 27, 29, 30, and p. 37 lines 3, 21 and 28:

peP

read:

peP

p. 37 line 1:

(lemma 7.1 and lemma 7.2 respectively) read:

CHAPTER I

I N T R O D U C T I O N

The theory of abelian groups* is universally known as a branch of algebra that has succeeded in obtaining several satisfactory structure theorems. During the last few decades research done in group theory has come to play an increasingly important role. The most important feature of the recent development is the prominent role played by homological algebra - as a tool it will be exploited to a great extent in our work.

The main topic of discussion of this thesis is the subject of neat

sub-groups together with the extension problem for sub-groups. The concept of

neatness, which is due to HONDA [7], is a natural generalization of

pure subgroups, a notion introduced by PRÜFER [9] in 1923. Priifer's discovery has led to many new ideas in group theory. Important is the notion of algebraic compactness: A group is said to be algebraically

compact if it is a direct summand of every group which contains it as a

pure subgroup.

One of the astonishing phenomena of group extensions is the discovery of HARRISON [3] that the extensions of ^ by 5 corresponding to pure exact sequences 0->y4-^G->5-+0 form a subgroup Pext(5,/I) of the group Ext(5, A) of all extensions of ^ by B. It follows that Pext(5, A) coincides with the first Ulm subgroup of Ext (5, A). In the same paper Harrison introduces the concept of a cotorsion group, defined as a group all of whose extensions by torsion free groups are splitting. Harrison adds the condition that a cotorsion group must be reduced. In order to be in conformity with the definition of algebraic compactness we shall, how-ever, omit this restriction. Owing to this convention the class of all algebraically compact groups is contained in the class of all cotorsion groups. The latter class has the property of containing the group Ext(fi, A), for arbitrary groups A and B.

A subgroup / / of a group G is called neat if the equality/;ƒ/=ƒ/n/jG holds for all prime numbers/?. Many of the properties of pure subgroups carry over to neat subgroups, but at the same time other well-known properties of pure subgroups are lost. Nevertheless, results in the

ing chapters will show that the study of neat subgroups may be very interesting and of some importance in studying the structure cf cotorsion groups. A complete survey of neat subgroups is given in §3.

In §4 the group Next (5, A) of neat extensions of A by B is introduced in the classical way, that is, by means of factor sets. It is shown that Next(5, A) coincides with the Frattini subgroup of E\t(B, A).

In §5 neat extensions are considered as short neat exact sequences and some homological properties of the functors Next (5, ) and Next( , A) are derived.

The structure of the group Next(B, A) is investigated in §6. It is proved that Next (5, A) is a cotorsion group for all groups A and B, and that the factor group Ext(5, ^)/Next(B, A) is always algebraically compact. Moreover, it is proved that Next(fi, A) inherits the property of algebraically compactness from Ext{B, A).

A group G for which there exist groups A and B such that G = Next (B, A)

will be called a Next-group. In §7 it is shown that every reduced torsion free cotorsion group is a Next-group. Every cotorsion group has there-fore a unique representation as a direct sum of three groups, one of which is a Next-group.

In §8 the notion of neat subgroups is generalized to that of n-neat

subgroups. This makes it possible to characterize the extensions of ^ by fi

which belong to the n-th Frattini subgroup of Ext (5, A).

In chaptei V the notion of algebraically compact groups is generalized to that of P,-groups, defined as groups all of whose pure extensions by torsion groups are splitting. It is proved that Ext (5, A) is algebraically compact for all groups B if and only if /4 is a P,-group. A necessary and sufficient condition for Ext (5, A) to be algebraically compact for all groups A, is also given.

§ 1 Preliminaries

Some of the definitions and results which will be used frequently later on are mentioned below, and unless otherwise stated they can all be found in [2]. G will always denote an additive abelian group.

Groups in which every element has finite order are called torsion

groups; those in which all the elements T^O are of infinite order are torsion free. By a p-group is meant a group in which the orders of the

elements are powers of the same prime number p. If there exists an upper bound for the orders of the elements of G then G is called a bounded

Every bounded group is a direct sum of (bounded) cyclic groups. Every torsion group is a direct sum of p-groups for various/? e P.

G is called a divisible group if nG = G for all neN. If H is a. divisible

subgroup of G then ƒ/ is a direct summand of G. Conversely, if /ƒ is a direct summand of every group containing it, then His a divisible group. Any group G can be embedded in a minimal divisible group. Between any two minimal divisible groups containing G there exists an iso-morphism leaving the elements of G fixed. Every group G has a unique decomposition G = dG QR where dG is the maximal divisible subgroup of G. If dG = 0 then G is said to be reduced.

Groups in which every element has square-free order are called

elementary. Every elementary /»-group is a direct sum of cyclic groups

of the same order p.

If A^ is a fixed subgroup of G then a K-high subgroup of G is a subgroup

H which is maximal with respect to the property HnK=0; if moreover G = H QKthen K is called an absolute direct summand of G.

A subgroup jy of G is called a pure subgroup if the equation nx =

heH (neN) is solvable in H whenever it is solvable in G; or equivalently, if nH=HnnG holds for all natural numbers n.

If /f is a subgroup of G such that the factor group GjH is torsion free, then H is pure in G. In particulai, fG is a pure subgroup of G.

The Frattini subgroup <P(G) of G is the intersection of all maximal subgroups of G.

$(G)=

The first Ulm subgroup of G is defined as

G' = HnG.

G/G^ is called the O-th Ulm factor of G, and is denoted by GQ.

If AT and L are groups then Hom(Ar, L) denotes the group of homomor-phisms of AT into L. If AT is the infinite cyclic group then Hom(A', L) = L, and if K is cyclic of finite order n then Hom{K, I,) = L[«]. In each of the following cases Hom (AT, L ) = 0 :

(i) K is torsion, L is torsion free; (ii) K is divisible, L is reduced;

(iii) Kis a/»-group, Z, is a ^-group, p?^^.

Let£ be a class of objects A,B,C,... together with a class of mappings (called morphisms) a, P, y,.... If to each pair (A, B) of objects there is

associated a set h.om(A, B) of morphisms, such that every a6hom(v4, B) has domain A and codomain (range) B, and if for every a e horn (.4, B),

Pehom{B, C) we have associated an element of hom(^, C), written as Pa, thenC is called a category if the following two axioms are satisfied:

(i) If a.e)xom{A, B), peh.om{B,C) and yehom(C, D) then

{yP)oi = y{Pa).

(ii) For each object B there exists a morphism Eg e hom (fl, B) such that Pe^ = p for all Pehom{B, C), and egoc = a for all

ixehom(A, B).

Let 9Ï denote the category of abelian groups. A function J : 9l-»-9I is called a covariant additive functor if, whenever OL: A-^B is a homo-morphism, there is a homomorphism r ( a ) : r ( / 4 ) - > r ( 5 ) such that

(i) r ( U ) = l r ( ^ , ;

(ii) T{p<x) =T(p)T((x) (y/heie p-.B-^C);

(iii) r ( a + y ) = r ( a ) + r ( y ) (where 7: ^ - > 5 ) .

A contravariant additive functor is defined similarly, except that

a: A-*B induces T{oi): T(B)-*T{A) and condition (ii) is replaced by

(ii)' T{Pa) = T{a)T{P) (where )S: 5 ^ C) (see [10] p. 207).

Let . . . ^ A _ / * ^ i A ^ A + i *

-be a sequence of groups and homomorphisms. This sequence is called

exact if the image of each homomorphism is equal to the kernel of the

next homomorphism. If 0-^A^B is exact then a is a monomorphism; if 5-+C->0 is exact then P is an epimorphism.

Let . ^ be a set of groups, let i ? be a set of homomorphisms between pairs of groups belonging to ^ , and let ^ be the set of all homomor-phisms obtained by composition of finitely many homomorhomomor-phisms in the set JS? ; thus JS? is completely determined by jSf. We say that the set J{ of groups and of homomorphisms forms a commutative diagram whenever the following condition is satisfied: If A,Be^ and f,ge £0" are such that ƒ : A-*B and g: A^B, then f=g. For example, if A, B, C, D are groups and a, p, y, 5, e are homomorphisms then the diagrams

C AUB D AU

n

cu

i'

are commutative if ot£=y, Pe=ö and p(x = 5y respectively. (The image of

aeA under the homomorphism a on /I is denoted by aa.) The Five Lemma

Consider the commutative diagram with exact rows of groups:

A^ -^ A2 ^ A2 -^ A^ ^ Aj

i"' i"' l"'

i"" i"'

JBI -^ JB2 -> B3 -> B4 -> B5

(i) If a2 and «4 are onto and a^ is one-to-one, then (x^ is onto. (ii) If a2 and «4 are one-to-one and a^ is onto, then aj is one-to-one.

§ 2 The extension problem

An extension G = {G, x) of a group ^4 by a group B is a pair consisting of a group G and a homomorphism x such that

0 ^ ^ - ^ G ^ B ^ O (2.1) is an exact sequence, where / may stand for the identity mapping.

We denote the elements of A and B respectively by 0, a,b,c,... and 0, a, P,y,... where 0 is the identity. A given extension (G, x) of Ahy B can be described in terms of coset representatives for the elements of B: For each a e B we select a fixed coset representative g{a) in the coset of G modulo A that corresponds to a under the epimorphism x- As representa-tive of O E B we choose the identity of A, that is g(0) = 0. It follows that

g{a) + g{p) = g{a + p) + ƒ ( « , p), ƒ(«, p)eA, oi, PeB

and

/ ( a , 0 ) = 0 forall « E B . (2.2)

Furthermore, the associative and the commutative laws in G imply the identities

ƒ (a + p, y) +f{a, p) = ƒ (a, jS -H y) +f{P, y) (2.3) and

f{a,p)=f{p,a) (2.4)

for all a, p, yeB.

A function ƒ : B x B ^ y 4 satisfying conditions (2.2), (2.3) and (2.4) is called a. factor set on B into A.

Conversely, given two groups A and B and a factor set ƒ on B into A, then the set G = B x ^4 forms a group under the operation

(a, a) -tiP,b) = (a-i-p,a + b +f(oc, P)), where

{(x,a) = {P,b) if and only if a = P and a = b.

The set of all elements of the form (0, a) is a subgroup A' of G and

A'^A under the mapping r : a ^ ( 0 , a). Furthermore, x:(a, a)-^a is a

homomorphism of G onto B with kernel A', that is

0 ^ ^ - ^ . G ^ B ^ O

is an exact sequence and thus (G, x) is an extension of A by B.

If we denote by F{B, A) the set of all possible factor sets on B into A, and if we define an operation

( ƒ + g) (a, P) =f{oi, P) + g{a, p), f geF{B, A), then F{B, A) is a group.

Two extensions G={G, x) and G' = (G', x') of Ahy B are said to be

equivalent, G~G', if there exists an isomorphism between them leaving

both A and B invariant. In other words, G~G' if and only if there exists an isomorphism d between them such that the following diagram with exact rows is commutative:

Q-^A^G ^ B ^ O

0 ^ A - ^ G ' X B - > 0

In terms of factor sets the definition of equivalence reads as follows: Two extensions G and G' of ^ by B determined by factor s e t s / a n d / ' respectively, are equivalent if and only if there exists a function a):B-*A such that

aj(0) = 0 (2.5) and

f(a, P) = / ( a , P) - / ' ( a , p) = «(a) -t- oi{p) - « ( a + p) (2.6) for alia,/?6B.

A factor set / for which there exists a function w.B^A satisfying (2.5) and (2.6) is called a transformation set.

The transformation sets form a subgroup T (B, A) of F (B, ^ ) . From (2.6) it follows that there is a one-to-one correspondence between the cosets of r ( B , A) in F{B, A) and the classes of equivalent extensions of

A by B. The group of non-equivalent extensions of ^ by B is therefore

defined as

Splitting extensions

The exact sequence (2.1) is called splitting exact if iA is a direct summand of G, and (G, x) is called a splitting extension of A by B.

The direct sum of A and B is an example of a splitting extension. Conversely, every splitting extension is equivalent to the direct sum of

A and B.

In terms of factor sets we have: An extension of ^4 by B determined by a factor s e t / i s splitting if and only if f eT{B, A). Thus every ex-tension of ^ by B splits (that is every exex-tension of ^ by B is equivalent to their direct sum) if and only if Ext(B, A) = 0. For example, if ^ is a divisible group then Ext(B, ^ ) = 0 for all groups B, and if B is a free group then Ext(B, ^4) = 0 for all groups A.

The functor Ext

It is well known that Ext(B, ) is a covariant additive functor and that Ext( , ^ ) is a contravariant additive functor (on the category of abelian groups). If

0-^A^B->C^O

is an exact sequence and G is any group, then we have the following two long exact sequences connecting Hom and Ext:

O^Hom(G, ^ ) ^ H o m ( G , B)->Hom(G, C ) ^

-> Ext(G, A) ^ Ext(G, B) ^ Ext(G, C) -^ 0 and

0 ^ Hom (C, G) ^ Hom (B, G) ->• Horn (/t, G)

^ E x t ( C , G ) ^ E x t ( B , G)-^Ext(.4, G)-+0. Let {Gj; iel} be any family of groups. Then

Ext(B, n G.) = n Ext(B, Gi)

iel iel

and

Ext(® G i , ^ ) ^ n E x t ( G j , ^ ) . 1 6 / iel

Pure extensions

The exact sequence (2.1) is called pure exact if iA i:> a pure subgroup of G, and (G, x) is called a pure extension of A by B.

Pure extensions were first studied by HARRISON [3]. He proved that the pure extensions of Ahy B form a subgroup Pext(B, A) of Ext(B, A) which coincides with the first Ulm subgroup of Ext(B, A). If B is torsion free then every extension of /I by B is a pure extension. Hence, i*" B is a

torsion free group then Pext(B, ^ ) = Ext(B, A) for all groups A.

We mention the following results from [3]: Pext(B, y4) = 0 for all groups A if and only if B is a direct sum of cyclic groups. Pext(B, A) = 0 for all groups B if and only if A = D @S, where D is a divisible group and 5 is a direct summand of a direct product of finite cylic groups.

Let

0 - > ^ ^ B ^ . C ^ 0

be a pure exact sequence and G any group. Then we have long exact sequences connecting Hom and Pext:

0 ^ Hom {G,A)^ Hom (G,B)^ Hom (G,C)^ ]

^ Pext (G, ^ ) -> Pext (G, B) -> Pext (G, C) ^ 0 and

0->Hom(C, G ) ^ H o m ( B , G)-^.Hom(^, G)-^

-* Pext(C, G) -y Pext(B, G) -> Pext(^, G) ->• 0.

Let {Gil iel} be any family of groups. Then

Pext(B, n G;) = n Pext(B, G,)

16/ iel

and

Pext(® Gi, ^ ) ^ n Pext(Gj, A).

iel iel

Algebraically compact and cotorsion groups

A group G is said to be algebraically compact if it is a direct summand

of every group which contains it as a pure subgroup. In other words, G is algebraically compact if and only if Pext (B, G) = 0 for all groups B. Using homological methods it is not difficult to prove that the conditions Pext(e/Z, G) = 0 and Ext(e, G ) = 0 imply that Pext(B, G)=0 for all groups B. Hence a group G is algebraically compact if and only if Pext(e/Z, G) = 0 and Ext(ö, G) = 0.

A group G is called cotorsion if all of its extensions by torsion free

groups are splitting, that is, if Ext(B, G) = 0 for all torsion free groups B. This is equivalent to the requirement that Ext(ö, G) = 0. Every alge-braically compact group is obviously cotorsion. A torsion free cotorsion group is algebraically compact.

Every cotorsion group G has a unique decomposition into a direct sum of three groups, G — D QA @B, where Z) is a divisible group, y4 is a reduced torsion free cotorsion group, and B is a reduced cotorsion group having no torsion free direct summand / O .

C H A P T E R II

NEAT S U B G R O U P S A N D E X T E N S I O N S

The concept of neatness of subgroups, which is due to HONDA [7], is a natural generalization of purity, and one could therefore expect that many of the properties of pure subgroups carry over to neat subgroups. In the following paragraph we give a survey of neat subgroups. After that the group of neat extensions is introduced and studied, the approach being the classical way, that is via factor sets.

§ 3 Neat subgroups

DEFINITION 3.1 A subgroup H of a group G is called a neat subgroup if

the equation px = heH, peP, is solvable in H whenever it is solvable in G. This is equivalent to the requirement pH = HnpG for allpeP, which is in turn equivalent to nH=HnnGfor all square-free natural numbers n.

Every pure subgroup of G is neat. Conversely, it is shown in [1] that if G={a} ©{6} where a and b are of order p^ and p respectively, peP, then the subgroup {pa+b} is neat but not pure in G.

To familiarize ourselves with the concept of neatness we state, among some new results, also results of Honda. It is understood that where no proofs are supplied they can be found in [1].

Neatness is a transitive property. Neatness is an inductive property; that is, the union of an ascending chain of neat subgroups is itself a neat subgroup. In torsion free groups neatness is equivalent to purity. If K is an arbitrary subgroup of a group G then every K-high subgroup of G is neat.

THEOREM 3.1 Let n be a square-free natural number and H a subgroup of

a group G such that H is a direct sum of cyclic groups of the same order n. Then the following statements are equivalent:

(a) H is a direct summand of G; (b) H is a pure subgroup of G; (c) H is a neat subgroup of G; (d) HnnG = 0.

Proof: That (a) implies (b) and (b) implies (c) are trivial. If H is neat

in G then mH=HnmG for all square-free natural numbers m. In par-ticular, if «ï = n then nH=0 so that HnnG=0, proving that (d) is a consequence of (c). To prove that (a) is a consequence of (d) we make use of the following lemma: If B is H-high in G and ifgeG such that for some

prime number p, pgeB then geH ® B (see [2] lemma 9.8). Now suppose

that (d) holds and let B be an //-high subgroup of G containing nG. The existence of such a subgroup is guaranteed by Zorn's lemma. We claim that G = H®B: Let n=PiP2.--Pr be the factorization of n into different prime numbers. If ^ e G then ngenG^B so that, by the quoted lemma, all 0fp2P3--Pr9,—,PiP2-Pi-iPi+i—Pr9,--,PiP2--Pr-i9 are in

H©B. Furthermore, there exist integers Aj, /l2,... A, with

1 = XiP2P3---Pr + ••• + ^iPlP2---Pi-lPi + l---Pr + •••

+ KPlP2--Pr-l

so that

9=^lP2P3--Pr9+-- + ^iPlP2---Pi-lPi + l-Pr9+-+ KPlP2--Pr-l9^ff ®B.

Thus GsH®B and since the converse inclusion always holds, H is a direct summand of G.

COROLLARY 3.2 If H is an elementary p-subgroup of G, peP, then the following statements are equivalent:

(a) H is a direct summand of G; (b) H is a pure subgroup of G; (c) H is a neat subgroup of G; (d) HnpG = 0.

Proof: In [2] it is proved that every elementary/>-group is a direct sum

of cyclic groups of the same order/», so that the condition of the previous theorem is fulfilled.

THEOREM 3.3 If H is a neat subgroup of G and if GjH is an elementary

^ group, then H is a direct summand of G.

Next we prove an analogue of the well known theorem that a bounded subgroup of a group G is pure if and only if it is a direct summand of G.

THEOREM 3.4 If H is a neat subgroup of G and if nH=0 where n is a

Proof: Let v: G-*Gl{H, nG} be the natural homomorphism. The factor

group G/{H, nG} is bounded and thus a direct sum of cyclic groups;

GI{H, nG} = © { x j Ae/t

where {x;^} is cyclic of order n^. For each X we choose x^^eG such that

vX;^=Xx. Then n^Xxe{H, nG} so that nxXx=h;^-\-ng;^, h^eH, gx^G. Since nxx = 0, nx\n (hence n^ is itself square-free for all A) and so

'1^ = " ^ ^ ^ 9x

I «A

Neatness of H in G implies the existence of h\eH with nxh\ = hx. If

yx=Xx-h\ then n^yx = ng,, and vj;i=X;^. Let K= {nG,... j ^ . . . } . We prove G = H®K:

(i) i f n ^ = 0 : l f x e / f n / s : then

AETI

while xeH implies vx = 0. Thus

S m^x;i = 0 XeA

and it follows that n;il»i/. But nxyx=ngxenG so that /«;j';ieMG. Therefore

^ = Z '"AJ'A + ngenG

XeA

and since ƒ/ is neat in G there exists heH with x=nh. But nH=0 so that x=n/i = 0.

(ii) G = {ƒƒ,/(:}: If fi(6G then

V0 = Z "^A^A • XeA Furthermore, since Z *"XXX = V Z "^A^A XeA we have Hence 0 = vg -V Z »ÏA}'A = v(g- Z "IA^A)-0 - Z "»A3'A6{W. nG} A d

and so

0 - Z "^A^A = ' i + "ö''> heH, g'eG. XeA

Thus

g = h + ng' + Z W^^A e {if,/C}.

XeA

If n is any prime power number and H a subgroup of G which is maximal disjoint from nG, then H is a direct summand of G ([2] theorem 27.7). Theorem 3.4 enables us to show that the same holds for all square-free natural numbers n.

THEOREM 3.5 If n is a square-free natural number then every nG-high

subgroup of a group G is a direct summand.

Proof: If H is, a subgroup of G which is maximal with respect to the

property HnnG—0, then H is neat in G, that is, mH=HnmG for all square-free natural numbers m. In particular, if m=n then nH=0 and the-orem 3.4 concludes the proof.

We pass on to two theorems which give necessary and sufficient conditions under which subgroups are neat.

THEOREM 3.6 A subgroup H of a group G is neat if and only if every coset

of G modulo H which has square-free order contains an element of the same order.

Proof: Let H be neat in G. If geG/H has order n,n a square-free

number, then ngeH for all geg. Hence there exists heH with nh=ng. Clearly, h — geg and has order n.

Conversely, hi heH and geG such ihai ng=h. Then n{g + H) = H and so the order of the coset g + H is n'\n. According to our assumption we may choose g'eg + H with order g'= n', but then ng' = 0 so that

n{g—g') = h where g—g'eH. We conclude that His neat in G.

If H is a subgroup of G we denote by n~^H (neN) the set of all geG

with ngeH. It follows that n~^H is a subgroup of G, containing H.

THEOREM 3.7 H is a neat subgroup of G if and only if H is a direct

summand ofn~^Hfor all square-free natural numbers n.

elementa-ry group, and theorem 3.3 implies that H, being neat in n~^H, is a direct summand of it.

Conversely, if ng = heH, geG, then gen'^H. If n~^H=H@K for some subgroup K of G, then g has a unique representation g=h'+k,

h'eH, keK. From h = ng = nh' + nk it follows that nk = 0 and h = nh',

so that i / is neat in G.

THEOREM 3.8 If H is a neat subgroup of G and A a subgroup of G con-tained in H, then H/A is neat in G/A. Conversely, if H/A is neat in G/A where A is a neat subgroup of G, then H is neat in G.

Proof: Let h + AeHjA and suppose there exists g + AeGjA such that p{g + A) = h-\-A, peP. Then pg = h + a, aeA, and neatness of H in G

implies that there exists h'eH withph' = h + a. But thenp{h'-{-A) = h + A, thus H/A is neat in G/A.

Now let H/A be neat in G/A where ^ is a neat subgroup of G contained in H. If pg=heH, geG,peP, thenp{g + A) = h+A from which it follows that h + A=p{h'-\-A) for some h' + AeH/A, or ph' = h + a, aeA. From

pg=h and ph'=h-\-a we have p(h'—g)=a, hence there exists a'e A such

that pa' = a. Thus h=ph'—a=p{h'—a') and h' — a'eH implies that H is neat in G.

Consequently, if /I is a neat subgroup of G then in the one-to-one correspondence between the subgroups of G/A and the subgroups of G containing A, neat subgroups correspond to neat subgroups.

In this paragraph we have treated one generalization of pure subgroups. We mention two more generalizations of this concept.

(a) p-Pure subgroups

Let J? be a fixed prime number. A subgroup H of a group G is called a

p-pure subgroup if the equaUty p'^H=Hr\p^G holds for all positive

integers k.

(b) y-Pure subgroups

Let 5^ be a subset of the set of all prime numbers and denote by &'* the semi-group generated by Sf. A subgroup Hofa group G is called 9'-pure

if nH^HnnG holds for all neS^*.

Many properties of pure subgroups remain true for/7-pure and 5^-pure subgroups. It is of course possible to generalize the concept of neatness to that of p-neatness or 6^-neatness, and to develop a whole theory

similar to that of neat subgroups. In stead of generalizing in this way, we shall generalize in a slightly different way to n-neat subgroups (see §8). These subgroups seem to play an important role in investigating certain subgroups of Ext(B, A).

§ 4 Neat extensions

DEFINITION 4.1 The exact sequence

0 - ^ A - ^ G ^ B ^ O

is called neat exact ifiA is a neat subgroup of G, and G = {G, x) is called a neat extension of A by B.

We shall not distinguish between the groups iA and A. Using the same notation as in §2, we let G be a neat extension of y4 by B a n d / a factor set on B into A, obtained after selecting coset representatives ^(a) in G. If;7a = 0, ;76P, then

p - i

P0(a) = »(«) + ö'(a) + . . . + 0(a) = Z / ( « . '«).

i = l

but this means that the equation

p-i.

P^ = Z ƒ (o'' '<=')

i = l

is solvable in G, hence it possesses a solution in A, P - i

Pff (a) = Z ƒ(°'''°') = P^' " ^ ^ •

1=1

Conversely, l e t / b e a factor set on B into A satisfying the condition

p - i

if pa. = 0, peP, then Z /(">'°')^P^> ('^•l)

i = l

and let G be the extension of ^4 by B constructed in §2. We proof that

A' is a neat subgroup of G: Let therefore (0, d)eA', peP and (a, b)eG

such that />(«, è) = (0, a). It follows that

p - i (pa, pb -I- Z ƒ («' '•«)) = (0' a ) , i = l so that/?a=0 and p - i 1 = 1

According to our assumption

p - i

Z ƒ (a, ice) = pa'

i = l

for some a'eA, and so a=pa where a=b + a'eA. Thus (0, a)={0,pa) =

p(0, a) proving that A' is a neat subgroup of G.

Consequently, every neat extension of ^4 by B determines a factor set

f on B into A which satisfies (4.1), and every factor s e t / o n B into A

satisfying (4.1) defines a neat extension of A by B.

Let F'{B, A) be the subgroup ofF{B, A) of factor sets/satisfying (4.1). We note that F'{B, A) contains T{B, A), for i f / 6 r ( B , A) t h e n / ( a , P) has for all a, PeB the form f{(x, P) = a){(x) + co{P)—co{cx.-\-P), where

co:B-yA is the function defined in §2. We thus have that

/ ( a , a) = co(a) + co(a) — cü(2a) / ( a , 2a) = cj(a) + cü(2a) — co(3a)

/ ( a , (P - 1) a) = (o{cc) -h (ü{(p - 1) a) - cü(pa). Therefore

p-i

Y, f(ix, ice) = pa)(a) - co(pa), 1 = 1

and ifpa = 0 then

p-i

Z ƒ (a, iot.) = pa)((x)epA.

1 = 1

Thus T{B, A) is a subgroup of F'{B, A).

Since two extensions G and G' of ^4 by B (defined by factor s e t s / a n d / ' respectively) are equivalent if and only if f—f'eT{B, A), we may

define the group of non-equivalent neat extensions of ^4 by B as Next(B, A) = F'(B, A)/T{B, A).

We observe that

(i) Next(B, A) is a subgroup of Ext(B, A) and since every pure extension is a neat extension, Pext(B, /4)s Next(B, A)^ Ext(B, A).

(ii) If either^ is divisible or Bis a free group then Ext (B, A)=Q, so that in such cases Pext(B, ^ ) = Next(B, A)=Q.

(iii) If A is an elementary p-group then Next(B, ^ ) = 0 for all groups B.

(iv) If B is an elementary p-group then Next(B, A) = 0 for all A. This is in accordance with a result of HAUPTFLEISCH [5] who proved that if B = Z ( n ) = { a } t h e n / e r ( B , A) if and only if

p - i

Z / ( a , i a ) 6 n ^ .

i = l

Consequently, if B is a cycUc group of prime order p then

F'(B, A) = T{B, A) so that Next(B, A) = 0.

(v) If B is torsion free then Pext(B, ^ ) = Ext(B, ^ ) and so Pext(B, .4) = Next(B, .4) = Ext(B, A), for all groups A.

The main theorem of this section states that the group of neat extensions

of Ahy B coincides with the Frattini subgroup of Ext(B, A). This being

an easy consequence of theorem 53.3 (i) of [2], which states that an extension G of y4 by B is divisible by a prime p if and only if A/pA is a direct summand of G/pA. But A/pA being a direct summand of G/pA for all p e P is equivalent to the neatness of A in G. Hence we have:

THEOREM 4.1

Next(B, A)= []p Ext(6, A) = ^(Ext(B, A)),

peP

where f?(Ext(B, A)) is the Frattini subgroup o/Ext(B, A).

When there is no danger of confusion we shall write

£'=Ext(B, ^ ) , N=Next{B,A), P = Pext{B,A). Several conclusions can now be drawn from the preceding theorem, we combine them in the following:

COROLLARY 4.2 (A) E is divisible if and only if N=E. Hence from

exercise 12 in §52 o / [ 2 ] it follows that Next (B,/I) = Ext (B,/I) if and only if A is divisible or B is torsion free.

(B) The following statements are equivalent: (i) N is a direct summand of E; (ii) A^ is pure in E;

(iii) A^ is neat in E; (iv) A'^ is divisible.

Proof:

(A) If E is divisible then pE=E for all peP so that

N= [) pE = E.

Conversely, if A'^=£'then (since N^pE for allpeP) E^pE, hence E=pE for a l l p e i ' .

(B) If (iii) holds then pN=NnpE for all peP. But N^pE so that

pN=NnpE=N, and N is indeed divisible. The rest is obvious. COROLLARY 4.3 IfN is divisible then N=P.

Proof: Using the fact that P= f] nE

neN

one can prove that if H is any divisible subgroup of E then it must be contained in P. Hence if N is divisible then N= P.

In chapter IV it will be proved that the condition N=P is also sufficient for A'^ to be divisible (theorem 9.5).

Theorem 4.1 enables us to compute Next(B, ^4) in case A and B are cyclic groups. Since every extension of /4 by a free group B splits, we shall take B finite cyclic. Moreover, by the direct sum properties of Next (see § 5) it is sufficient to consider B as a cyclic group of prime power order. We distinguish between the cases where A is infinite and finite respectively.

(a) A is an infinite cyclic group

Suppose B is cyclic of order p", peP and « > 1 an integer (if « = 1 then Next(B, ^ ) = 0). It is well known that Ext(B, A) = A/p''A, which implies that Ext(B, y4) = Z(p") if A is infinite cyclic. Hence

Next(B, /I) = f) p Ext(B, A) = pZ(p") = Z(p''~i).

peP

(b) A is a finite cyclic group

We may consider ^4 as a cyclic group of prime power order. If ^ = Z(q'"),

B=Z{p'') where p^q are prime numbers then Next(B, ^ ) = 0. Let

therefore A=Z(p'"), B = Z(p'') where m,n'^2 (if either m=l or n=l then Next(B, ^ ) = 0 ) . It follows that

Ext(B, A) = Z{p'")/p-Z(p'") = Z(p-")/Z(p'"-') = Zif),

where r=min {m,n). Consequently

Next(B, A)= (\p Ext(B, A) = pZ{p'^ = Z(p"'').

HAUPTFLEISCH [6] has proved that, with A and B as above, there are

p' — p'~^ cychc extensions of A by B. Since a cyclic p-group contains no

non-tri vial neat subgroups, we conclude that the neat extensions of

Z{p'") by Z{p'') are exactly those extensions which are non-cyclic.

The following theorem is a restatement of corollary 4.2 (A).

THEOREM 4.4 (i) Next(B, A) = Ext(B, A) for all groups B if and only if

A is a divisible group.

(ii) Next(B, y4) = Ext(B, A) for all groups A if and only if B is a torsion

free group.

THEOREM 4.5 (i) Next(B, A)=Ofor all groups B if and only if

A==D®YlT^,

peP

where D is divisible and pTp = 0.

(ii) Next(B, A) = Qfor all groups A if and only ifB = F®S where F is

free and S is a direct sum of elementary cyclic groups.

Proof: The proof of (i) can be found in [4] (lemma 4). As for (ii),

we know from theorem 3.3 that if B is a direct sum of elementary cychc groups then Next(B, .4)=0. Consequently the direct sum properties of Next(B,/4) (see §5) imply that Next(B, ^ ) = 0 whenever B has the mentioned structure. Conversely, if Next(B, y4) = 0 for all groups A then also Pext(B, ^ ) = 0. Hence B is a direct sum of cyclic groups,

B=F®S where F is the direct sum of all infinite cyclic groups and S

is a direct sum of cyclic p-groups. The example preceding theorem 4.4 shows that every direct summand of S must be elementary.

THEOREM 4.6 The Frattini subgroup of a group G vanishes if and only if

G is isomorphic to a subgroup of

n

z(p).

peP

In particular, Next(B, A)=0 if and only j / E x t ( B , A) is isomorphic to a subgroup of

n

2(p).

peP

Proof: If $ ( G ) = 0 then consider the homomorphism

(p:G-^Y\ G/pG

defined by

<pg = {..., g+pG, ...),geG. lf{...,g+pG, ...)=Othen

gef] pG = 0,

peP

hence <p is a monomorphism. Furthermore, since pG is a maximal subgroup of G we have G/pG^Z{p). The converse is clear and the theorem is proved.

C H A P T E R III

H O M O L O G I C A L M E T H O D S

In the preceding section the method of factor sets was used to discuss

the group of neat extensions. We now turn to homological methods, and in §5 we shall closely follow the approach of [8] of extensions as short exact sequences. Some homological properties of Next are derived; these are used in §6 where the structure of the group Next(B, A) is discussed. In §7 it is investigated what groups may be considered as groups of neat extensions.

§ 5 Neat extensions as short exact sequences

DEFINITION 5.1 A neat extension of a group A by a group B is a short

neat exact sequence

N: O^A^G-^B^O (5.1)

If 91 denotes the class of all short neat exact sequences, then 91 becomes a category if we define a morphism between two objects A'^ and A^' of 91 as a triple (p = {a., y, p) of group homomorphisms such that the follow-ing diagram commutes:

JV: Q-*A A G ^B - > 0

N': O^A' ^G'^B'-^O (5.2)

In accordance with the definition in §4 we shall say that the extensions

A'^ and A'^' of ^4 by B are equivalent, N~N', if the diagram (5.2) with

A=A', B=B', a = l ^ and P=IB is commutative.

THEOREM 5.1 Given a neat extension NofAbyB and a homomorphism p

of a group B' into B, then there is a neat extension Np of A by B' (which is unique up to equivalence),

NP: 0^A!^G'^B'^0,

and a homomorphism y:G'^G such that the following diagram commutes:

NP: O-^A^G'^B'^O

On the other hand, ifp is a homomorphism ofB' onto B with Ker(j9) neat in

B', and if N is an extension of A by B such that the induced extension NP

of A by B' is a neat extension, then N is itself a neat extension.

Proof: Define G' as the subgroup of G ®B' consisting of all those

elements (g, b'), geG, b'eB' for which vg=pb'; define

y :G'-^G ; y{g,b') = g, n':A-*G'; p.'a = (fia,0),

and

v ' : G ' ^ B ' ; v'(g,b')=b', aeA, b'eB', geG. Then

O - ^ ^ ^ G ' X B ' ^ O

is an exact sequence and the above diagram is commutative (cf. [8]). It remains to prove that the neatness of nA in G implies neatness of

p'A in G'. Let therefore p{g,b')={fia,0),peP,{g,b')eG', {na,0)efi'A. Then pg=na=piia' for some iia'e^A (the last equality since nA is neat

in G), and so (//a, 0)=p{na', 0).

On the other hand, let P be an epimorphism (hence the Five Lemma implies that y too is an epimorphism) and let Ker(^) be a neat subgroup of B'. Suppose that NP is a neat extension of A by B'. We wish to prove

fiA neat in G. Let therefore pg^ = ^a, geG, fiaefiA, peP. Since y is epic, 9 — 19' for some g'eG'. Furthermore, the commutativity of the left hand

square in the above diagram implies na—yn'a. Thus pyg' = yn'a and it follows that

pg' = n'a + k, (*)

where keKer(y). We observe that if yk = 0 then also vyk = 0, and since the right hand square of the above diagram is commutative, Pv'k=0. Thus if ^6Ker()') then v'A:6Ker(jS). Applying v' to (*) we have

pv'g' = v'fi'a + v'k = v'k.

But this means that the equation px= v'keKer(p) is solvable in B, and our assumption assures the existence of k'eKer(p) mthpk' = v'k. Since v' is epic there exists g'eG' with v'g" = k'. Hence pv'g'' = v'k and it follows that pgi'' — A:6Ker(v') = Imp'. Letpg" — k = n'a'ep'A. Then

pg" = H'a'+k, (**)

where Pv'g'' = 0. It follows from (*) and (**) that

p{g' -g") = ^l'a-n'a',

and neatness of p'A in G' implies the existence of n'a"e^'A such that

Hence

yp(p.'a'') = yn'a — yn'a'

or

p{pa") = na— yp'a'.

We prove that yp.'a'epp.A: From (**) we have yn'a'= —yk+pyg'^pyg". But jSv'öf"=0, hence vyg''=0, that is, yg''eKer(v)=lmp. Thus yp.'a! —

P19"=p(f^a) for some paefiA, and so

p{y-a") = fia — yp'a' = fia — ppd or p(//a" + fia) = pa.

Consequently fiA is a neat subgroup of G.

The uniqueness of NP implies the equivalences

NIB~N and N{PP')^{Np)P', (5.3)

where iS:B'^B,j8':B"^B'.

Returning to factor sets we show how a factor set for the extension

Np can be constructed from a given factor set for the extension A^. THEOREM 5.2 Let P:B'-yB be a homomorphism. If f: BxB-yA is a

factor set on B into A satisfying (4.1) then f':B'xB'-*A, defined by f'{bi b2)=f{Pbi pb2) for all * ; , * 2 e B ' , is a factor set on B' into A

which also satisfies (4.1). Moreover, if f is a factor set for N then f' is a factor set for Np.

Proof: If pb' = 0, peP, b'eB', then also ppb' = 0. Assuming that /

satisfies (4.1) we therefore have

'Y.f{pb',ipb')epA.

1 = 1

Hence

'Y.f'{b',ib')epA.

i = l

Let {g{b); beB} be a set of coset representatives in G for the elements of B, and l e t / b e the factor set it determines. For b'eB' we choose as the corresponding coset representative in G' the element {g (Jib'), b'). It follows that the factor set determined by the representatives

{{giPb'),b'); b'eB'}

is exactly/'. This proves the theorem.

group B' into B we let a be a homomorphism of A into a group A'. We establish a result which displays a remarkable duality with theorem 5.1.

THEOREM 5.3 Given a neat extension N of A by B and a homomorphism

<x:A^A', then there is a neat extension uN of A' by B (which is unique up to equivalence),

aN: 0 ^ ^ ' ^ G ' X B ^ O ,

and a homomorphism y:G^G' such that the following diagram commutes: N: 0-*A -l^G ^B^O

r i \ 11'"

aN: O^A' !^G' ^B^O

On the other hand, let a. be a monomorphism of A into A' whose image is a neat subgroup of A'. If N is an extension of A by B such that the induced extension aN of A' by B is a neat extension, then N is itself a neat extension. Proof: Let H he the subgroup of ^ ' @ G consisting of all elements of the

form {aa, —pa), aeA. Define G'= {A' @G)/H,

y:G-yG'; yg={Q,g) + H, H':A'-^G'; p'a' ={a',0)-\-H,

and

v ' : G ' - * B ; v'[(a', g)-\-H^ = vg, a'eA', geG. Then

0-*A'^G'^B-^0

is an exact sequence and the above diagram commutes (cf. [8]). We now prove that neatness of fiA in G implies neatness of n'A' in G': If

p [{a', g)-\-H'\ = (a', 0) + H, peP, (a', g) + HeG', {a', 0) -1- Hep'A',

then {pa',pg)-\-{aa, —pa) = {a',0) for some (aa, —na)eH. Hence a'=

pa'+aa and pg=pa=ppa, fiaepA (the last equahty since p.A is neat

inG).

Thus (a', 0) = (pa' 4- a a, ppa —pa)

= (pa' + aa+ paa — paa, ppa — pa)

=p(a'+ aa,0) + (a(a~pa), —p(a—pa)).

Since (a(a—pa), —p(a—pa))eH we have

(a', 0) + H = p[(a' + aa, 0) -I- H] epp'A',

On the other hand, let a be a monomorphism of ^4 into A' and let aA be a neat subgroup of A'. Suppose that p'A' is a neat subgroup of G'. We prove that ^lA is a neat subgroup of G: If pg=pa, peP, p.aepA, geG, then pyg = yp.a=n'aa. Since p'A' is neat in G' there exists p'a'ep'A'

mthpn'a'=p'aa, hence p a ' = aa and neatness of ay4 in A' impliesp(aa) =

aa for some a a e a ^ . Since a is a monomorphism we have pa = a and application of p yields pfia=pa, that is, pA is neat a subgroup of G.

The uniqueness of aA'^ implies the equivalences

l^N~N and (a'a) N'^ a'(aN), (5.4)

where a:A-*A', a':A'^A''.

THEOREM 5.4 Let N be a neat extension of A by B and let a: A^ A' be a

homomorphism. Iff:BxB-*A is a factor set on B into A which satisfies

(4.1) then f' = af:BxB^A' is a factor set on B into A', also satisfying (4.1). Moreover, if f is a factor set for N thenf' = afis a factor set for aN.

Proof: lfpb = 0,peP,beB, then

p - i

Z f(b, ib) = paepA.

i = l

Hence

Z f'(b, ib) = Z («ƒ) (b, ib) = a (pa) = p ( a a ) 6 p ^ ' .

i = i i = i

Let {g(b); beB} he a set of coset representatives in G for the elements of B and l e t / b e the factor set it determines. In G' we choose (0, g(b)) + H as representative of beB. It follows that the factor set determined by the representatives {(0, g(b)) + H; beB} is e x a c t l y / ' = a/.

DEFINITION 5.2 If fi:A^B and p':A'-*B' are homomorphisms we define

p®p':A®A'^B®B' by p®p.'(a,a') = (pa, p'a'),(a,a')eA ®A'. The diagonal map Aa:G^G ®G of agroup G is defined by AQg=(g, g), geG; and the codiagonal map VQ:G ®G-yG by VQ(g, g')=g+g', g, g'eG. DEFINITION 5.3 Let

N : 0->A^G A B ^ O

and

JV': O ^ ^ ^ G ' X B - > 0

be two neat extensions of A by B. Their direct sum is defined as the extension

It follows that N ®N' is a neat extension of A ®A by B ©B. We note

that if f and f' are factor sets of N and N' respectively, then f+f':(B ®B)x(B®B)-^A ®A, defined by

if+f') ((*1, ^2), (.b'u b'2)) = (f(b„ b',), f'(b2, b'2)),

(bi,b2),(b'i,b'2)eB © B, is a factor set for N ®N'.

Proof: Let {g(b); beB} and {g'(b); beB} he fixed coset representatives

in G and G' respectively, and l e t / a n d / ' be the corresponding factor sets defined by these representatives. As representative of (èj, b2)eB ®B we choose (g(bj), g'(b2))eG ®G'. lf(b'i, b'2) is another element of B ©B then

i9(b,),g'(b2)) + (g(b',),g'(b'2)) =

= {g(bi) + g(b'i),g'(b2) + g'(b'2))

= {9(bi + b',) +f(b„ b',), g'(b2 + b'2) +f'(b2, b'2))

= i9(bi + b',), g'(b2 + ^2)) + {f(bu b',),f'(b2 b'2)).

It follows that the factor set determined by the representatives

{{9(bi),g'(b2)); (b„b2)eB®B}

is exactly f+f'.

It remains to prove t h a t / - ! - / ' satisfies condition (4.1). Let therefore

p(bi, b2) = 0 for some prime number p and (b^, b2)eB ®B. Then pb^ = pb2=0, and hence p - i p-i Y, f(bi,ibi)epA and ^ / ' ( ^ 2 , ' 2 ) 6 ^ ^ -i=i i = l Now Z (ƒ + ƒ') ((fci,i'2),'(fel, M ) i = l = ''J:{f(buib,),f'(b2,ib2)) 1 = 1 = ( Z /(fei.'feiXZ f'(b2,ib2))ep(A®A). i = l i = l

Our assertion is proved.

We have seen in §4 that the neat extensions of ^4 by B form a group Next(B, A), where the operation was defined in terms of factor sets. The above two definitions enable us to define an operation in the language of short exact sequences (see definition 5.4 below). After that we prove

(theorem 5.5) that these two definitions are in perfect accordance with each other.

DEFINITION 5.4 The sum of two neat extensions N and N' of A by B is the

extension N+N' = V^(N®N') Ag.

It follows from theorems 5.1 and 5.3 that N-{-N' is again a neat extension.

THEOREM 5.5 If Nf denotes the neat extension of A by B determined by

the factor set f then Nf+j;~Nf-\-Nj:.

Proof: By definition, Nf+j-, has f+f' as a factor set while Nf + Nf.

has V ^ ( / - f - / ' ) ' , defined by V ^ ( / + / ' ) ' (^i, ^2) = V ^ ( / + / ' ) ( A A , A A ) , (éi,Z)2)eB©B, as a factor set. We assert that V ^ ( / + / ' ) ' = / + / ' : If èj and ^2 are arbitrary elements of B then

V x ( / + / ' ) ' (bu b2) = V ^ ( / + / ' ) ((b„ b,), (b2, b2))

= yA(f(b„b2),f'(b„b2))

= ƒ ( ^ , * 2 ) + ƒ ' ( ^ , * 2 ) = ( / + ƒ ' )

(*1,*2)-For neat extensions N and N' of A by B and homomorphisms

a, a'eHom(/4, A') and P, P'eUom(B', B)

the following equivalences hold:

a(N + N') ~ aN + aN'; (5.5) (N+N')P'^NP + N'p; • (5.6)

(a + a')N~aN + a'N; (5.7)

and

N(p + P')~NP + Np'. (5.8)

The others being more or less trivial, we shall prove only (5.8).

Let / be a factor set determining TV. Then theorem 5.2 implies that

Np, NP' and N(P + P') have factor sets/1,/2 and/3 respectively, where fiib'y, b'2)=f(Pb'„ Pb'2),f2(b'u b'2) =f(P'b'„ p'b'2)

and

/3(fe'i, b'2) =f(pb', + P'b'u Pb'2 + P'b'2), b'„ b'2eB'.

Since Nfi + Nf^~Nj-^^.f^, our assertion will follow if we can prove

Nf^+f^~Nj-^, which will be the case if/ -i-fz—fs is a transformation set.

In order to prove that / +/2 —f3 is a transformation set we need the following technical lemma.

LEMMA 5.6 Let f be a factor set on B into A. Then

fibu b2) +/(è3, M -f{bi + ^3, b2 + b^) =

ƒ (fei, fta) + ƒ (*2, M -fibi + b2, b3 + M

forallb;eB, /=1,2, 3, 4.

Proof: Applying the identity (2.3) twice we see that fibi+b3,b2-\-b;) =

= f{bu b2+b3+b^) +f(b3, b2+b^)-f(b„ 63) =f{bi+b2,b3+b^)+f(b„b2)-f(b2,b3 + b,)

+f(b3,b2+b^)-f(b„b3).

But (2.3) and (2.4) together imply that

ƒ (*3, ^2 + ^4) -f{b2, 63+^4) =7(^3, M -/(fe2, ^4) so that f(b,+b3,b2 + b^) = = f(b,+b2, b3+b^) +f(b„ b2) +f(b3, b^) -f(b„ b3) -fib2,b,). Hence f(bi,b3)+f(b2,b^)-f(b,+b2,b3 + b^) = fipu b2) +f(b3, M -fibi +b3, b2 + b^).

The lemma is proved.

Returning to (5.8) we let b'l and b'2 be arbitrary elements of B'. Then

(/l+/2-/3)(*'l,fe'2) =

fl{b'l,b'2)+f2(b'ub'2)-f3(b'ub'2)

= f(Pb'„ Pb'2) +f(P'b'„ P'b'2) -f(Pb',+P'b'„ Pb'2+P'b'2).

Applying lemma 5.6 we have

{f,+f2-fz){b'r,b'2) =

=f(pb'„ P'b',) + f (pb'2, P'b'2) -f(pb', +pb'2, P'b',+P'b'2)

= 0)(b',) + 0)(b'2)-0}(b',+b'2),

where co defined by a)(b')=f(pb', P'b') for all b'eB', is a function on B' into A satisfying cü(0) = 0. Hence / 4-/2—/s is a transformation set andsoA^ƒ3~Afƒ,-HAfƒ^.

From (5.5) it follows that the mapping

a*: Next(B, ^)-vNext(B, ^ ' ) ; a^N=aN, is a group homomorphism. Thus a e Horn (^, A') induces

Furthermore, equivalences (5.4) and (5.7) show that Next(B, ) is a covariant additive functor.

Similarly, peHom(B', B) induces a homomorphism

P*: N e x t ( B , / ( ) ^ N e x t ( B ' , ^ ) ; p*N = Np,

and from (5.3) and (5.8) it follows that Next( , ^4) is a contravariant additive functor.

Finally, equivalences (5.7) and (5.8) respectively imply that A^*: H o m ( ^ , ^ ' ) ^ N e x t ( B , ^ ' ) ; N*a = aN and

A^* : Hom(B', B) -^ Next(B', A); N^p = Np are homomorphisms.

Exact sequences for Next

If a is a homomorphism of A into B and if K is an arbitrary group, then a.':r\-^ari (with rie\{om(K, A)) is a homomorphism of Hom(A^, A) into H o m ( ^ , B), and a'':x^X<^ (with x6Hom(B, K)) is a homomorphism of Hom(B, A:) into H o m ( ^ , K).

It is well known that a short exact sequence

Q^A^B^C-*0 (5.9)

gives rise to long exact sequences

0-> Hom (/s:,^)-.-Horn (/s:,B)^ Hom (/i:, C)^

-^ Ext (K, A) -* Ext (K, B) -^ Ext (K, C) -* 0 (5.10)

and

0 -^ Hom(C, K) -y Hom(B, K) -^ Hom(A, K) -^

-yExt(C, K)^Ext(B, K)-yExt(A, K)-^0 (5.11)

for an arbitrary group K.

The question arises as to whether (5.9) yields exact sequences similar to (5.10) and (5.11) for Next. This is however not true unless a further restriction is placed on (5.9).

THEOREM 5.7 If the sequence

N: 0-*A^B^C->-0

is neat exact, and if K is any group, then the following sequences are exact:

0 -> Hom (K, A) X Hom (K, B) ^ Hom (K, C) -^

and

O -^ Hom (C, K) £; Hom (B, K) '^ Hom (^, X) ^

Z Next (C, -K) ^ Next (B, X) X Next (yl, K) ^ 0. (5.11)' Furthermore, a'Hom(Ar, A) and P''Hom(C, K) are neat subgroups of

Hom(X, B) and Hom(B, K) respectively.

Proof: In the preceding discussion of Next we have prepared the way

for a proof of the exactness of (5.10)' and (5.11)' which is similar to that of theorem 51.3 of [2]. The proof of the exactness is therefore omitted.

In order to proof that a'Hom(X, A) is a neat subgroup of Hom(X, B), we let pr\ = a'(p, where p is a prime number, rieYiom(K, B) and

(peY{om(K, A). But then pr\ = a'(p:K-*aA, and hence ri:K-^p~^(aA) = aA @H, for some subgroup H of B (see theorem 3.7). If keK and if rjk — aa + h, aaeaA, heH, we define a new homomorphism x'K^aA by xk = aa. Now xeHom(X, B) and so there must exist il/eiiom(K, A) such

that a'\lf = x- It follows that p(a'\l/)=px=pri = a'cp, so that a'Hom(A', A) is a neat subgroup of Hom(X, B).

Next we prove that j8"Hom(C, K) is neat in Hom(B, K): Let therefore

pri = P"(p, w i t h p e P , >;eHom(B, K) and (pe¥lom(C, K). We observe that aAS:Ker(P"(p), for if aaeaA then P"(paa = (pPaa = 0. Also, pKer(p/j)E

Ker(>j), for if xeKer(pf;) then ri(px) = (pri) x = 0. Thus p(a^)cKer(/;). Since a^ is pure in B, there exists a subgroup / / of B such that B/p(aA) —

aA/p(aA) ®H/p(aA). Every beB therefore has the form b = aa + h+paa', heH, aa, aa'eaA. Define a new homomorphism x'P^K by xb = 0 if beaA and xb = *]b if beH. If c = pbeC, define a new mapping i// of C

into X b y il/c=xb. Since a ^ £ K e r ( x ) , "A is well defined. It follows that

p(P"il,)=pX=pri=:p''(p. Hence P"Hom(C, K) is neat in Hom(B, K). THEOREM 5.8 Let {Ai; iel} be a family of groups. For any group K,

Next(© Ai, X) ^ n Next(^,., K) iel iel and

Next(K, n ^i) = n Next(X, A,).

iel iel Proof: We know that

Ext(® Ai, K) S Y[^xt(Ai, K) iel iel and

Ext(X,nA) = nExt(X,^i). iel iel

Since Next(B, A) is the Frattini subgroup of Ext(B, A), our theorem will follow if we can prove that, for any family of groups {G,-; iel},

^(UGi) = U^(Gi).

iel iel

But this follows from the equality

n piUGi)=nif] pGi).

peP iel iel peP

The theorem is proved.

§ 6 Some properties of Next

It is known that the group of all extensions of .4 by B is (for arbitrary groups A and B) a cotorsion group. Although subgroups of cotorsion groups are in general not cotorsion, we can prove that Next(B, A) is always a cotorsion group.

THEOREM 6.1 Next(B, A) is for all groups A and B a cotorsion group.

Proof: Let A and B be arbitrary groups. The factor group Ext(B, A)/

Next(B, A) is reduced. In fact, we even have that its Frattini sub-group is zero. To prove that Next(B, A) is cotorsion we show that Ext(e,Next(B,.4))=0.

From the exact sequence

0->Next(B, A)-* Ext(B, A)->• Ext(B, ^)/Next(B, ^)-> 0 we have the exactness of

Hom(e, Ext(B, ^)/Next(B, ^))-^'Ext(e, Next(B, A))^

-yExt(Q,Ext(B,A)).

Since Ext(B, A)/Next(B, A) is reduced the first group vanishes, and since Ext(B, A) is cotorsion, Ext(ö, Ext(B, A)) = 0. Hence Ext(e, Next(B, A)) = 0, as stated.

The next question that arises is under what conditions will Next(B, A) be algebraically compact; that is, under what conditions will the purity of Next(B, A) in a group G imply that Next(B, A) is a direct summand of G. Subgroups of algebraically compact groups are in general not algebraically compact, but we prove

THEOREM 6.2 IfExt(B, A) is algebraically compact, then so is Next(B, A).

algebraically compact if and only if its reduced part is, we may assume that Next(B, ^ ) is reduced. This implies that Ext(B, ^4) is reduced. Hence Pext(B, A), being the first Ulm subgroup of Ext(B, A), vanishes (see the remark following corollary 38.2 in [2]). But if the first Ulm subgroup of Ext(B, A) vanishes then so does the first Ulm subgroup of Next(B, A). We have proved that the first Ulm subgroup of the reduced cotorsion group Next(B, A) is zero, and so proposition 54.2 of [2] implies that Next(B, A) is algebraically compact.

Concerning the factor group Ext(B, y4)/Next(B, A) we have

THEOREM 6.3 Ext(B, ^)/Next(B, A) is for all groups A and B

algebrai-cally compact.

Proof: The Frattini subgroup (and hence the first Ulm subgroup) of

the factor group Ext(B, y4)/Next(B, A) is zero. Furthermore, since a homomorphic image of a cotorsion group is cotorsion, we have that Ext(B, .4)/Next(B, ^ ) is a reduced cotorsion group whose first Ulm subgroup vanishes. Hence it is algebraically compact.

For the sake of completeness we state two theorems concerning Pext. The proofs are much the same as those of theorems 6.2 and 6.3 respec-tively.

THEOREM 6.4 If Next (B, A) is algebraically compact, thenso is Pext(B, A).

THEOREM 6.5 Next(B, .4)/Pext(B, A) is for all groups A and B

alge-braically compact.

Let us note that theorems 6.1, 6.2 and 6.3 can be stated in a more general form, namely:

THEOREM 6.1' If G is a cotorsion group, then the Frattini subgroup $(G) ofG is again cotorsion.

THEOREM 6.2' If G is an algebraically compact group, then so is <P(G).

THEOREM 6.3' If G is a cotorsion group, then G/4>(G) is algebraically

compact.

We proceed to prove a few isomorphisms concerning Next, and for the sake of completeness, the analogues for Ext and Pext will in some cases be stated as well.

LEMMA 6.6 Let Abe a cotorsion group. Then (i) Ext (B, ^ ) ^ E x t (tB,A); (ii) Next(B, A) s Next(/B, A);

and

(iii) Pext (B, A) S Pext (?B, A).

Proof: The neat exact sequence 0->/B->B->B//B-»0 imphes the

ex-actness of the sequence

Next (B/fB, ^)->Next(B, v4)^Next(?B, ^ ) - > 0 .

Since A is cotorsion, Next(B//B, A)=0. Hence Next(B, A)^Next(tB, A).

LEMMA 6.7 Let A be a torsion free group. Then Pext(B, A) = 0 for all

torsion groups B.

Proof: If A is torsion free then Ext(B, A) is algebraically compact

whatever B is. Hence Pext(B, ^ ) , being the first Ulm subgroup of Ext(B, A), is divisible ([2] exercise 8, §38). If however, B is a torsion group, then Ext(B, A) is reduced ([2] lemma 58.4). Thus Pext(B, A)=0, as stated.

COROLLARY 6.8 Let A be a torsion free group. Then A is algebraically

compact if and only if it is cotorsion.

Proof:* Suppose that y4 is a torsion free cotorsion group. If B is an

arbitrary group then we have (by the previous two lemmas) that Pext(B, A) ^ Pext(?B, A) = 0.

Hence A is algebraically compact. The converse is obvious.

THEOREM 6.9 Let Abe a torsion free group. Then (i) Ext (B,^)^Ext (tB,A)®Ext(B/tB,A); (ii) Next (B, A) S Next (fB, A) © Next (B/tB, A);

and

(iii) Pext (B, A) ^ Pext (B/tB, A), for arbitrary B.

Proof: The sequence O-^tB-yB^B/tB^O is neat exact. Hence

Hom(?B, ^)^Next(B/rB, ^ ) ^ N e x t ( B , ^)-^Next(/B, A)^0 is exact. The first group is zero since tB is torsion and A is torsion free, and the second group is by corollary 4.2 divisible. Thus the exact sequence

0^Next(B/?B, A)-^Next(B, A)-^Next(/B, A)^0

is splitting and we have

Next(B, A) s Next(rB, A) © Next(B/rB, A).

THEOREM 6.10 IfB is a torsion group then the sequences

(i) 0->Ext (B,/^)-*Ext (B, ^ ) - » E x t (B,A/tA)-*0; (ii) 0 -* Next(B, tA) ^ Next(B, A) -^ Next(B, A/tA) -> 0;

and

(iii) 0 -> Pext (B, tA) -»Pext (B, ^ ) ^ 0,

are exact for arbitrary A. Moreover, Ext(B, tA) is a pure subgroup of

Ext(B, A) and Next(B, tA) is a pure subgroup of Next(B, A). If A is

algebraically compact then the first two sequences are splitting.

Proof: We prove the exactness of (ii) first. The neat exact sequence 0^tA^A^A/tA-*0 implies the exactness of

Hom(B, A/tA) -^ Next(B, tA) ->• Next(B, A) -^

-y'Next(B,A/tA)^0.

Since B is torsion, Hom(B, A/tA) = 0, and the exactness of (ii) follows. The exactness of (i) can be verified similarly. To prove that Pext(B, tA) = Pext(B, A) we repeat the above argument to obtain the exact sequence

0 = Hom(B, A/tA) ->• Pext(B, tA) ->• Pext(B, A) ->•

--yPext(B,A/tA)^0.

From lemma 6.7 we have Pext(B, A/tA) = 0. Hence Pext(B, M ) ^ Pext(B, A). Next we establish the purity of Next(B, tA) in Next(B, A). L e t / : B x B - > M and g:BxB^A he factor sets representing elements of Next(B, tA) and Next(B, A) respectively, and suppose that ng=f, n a positive integer. Then ng(bi,b2)=f(bi,b2) for all b,,b2eB. But ƒ(*!, b2)etA, therefore g(bi, b2)etA for all b,, b2eB. Thus g represents

an element of Next(B, tA), proving that Next(B, tA) is pure in Next(B, A). Now let A be an algebraically compact group. We shall prove that (ii) is splitting by proving that Next(B, tA) is algebraically compact. Since Ext(B, A) is algebraically compact whenever A is algebraically compact, it follows from theorem 6.2 that Next(B, y4) is algebraically compact, and so the following lemma completes the proof.

LEMMA 6.11 Let B be a torsion group. Then Next(B, A) is algebraically

compact if and only //Next(B, tA) is algebraically compact. The same is true for Ext, and obviously also for Pext.

o -* Next(B, tA) -^ Next(B, A) -^ Next(B, A/tA) -^ O

is pure exact, and since A/tA is torsion free the last group is algebraically compact. Furthermore, Next(B, A/tA) is reduced, and so the first and the last groups in the following exact sequence are zero

Hom(Z(p°°), Next(B, ^/M))->Pext(Z(p""), Next(B, tA)) -> -^Pext(Z(p'"), Next(B, ^))^Pext(Z(p°°), Next(B, A/tA)). Hence Pext(Z(p°°), Next(B, ^ ) ) s P e x t ( Z ( p ' " ) , Next(B,/y4)), and the lemma follows.

§ 7 Groups that are groups of neat extensions

We know from [2] that every reduced cotorsion group G can be considered as the group of extensions of a suitable group ^4 by a suitable group B. In fact, if G is a reduced cotorsion group then G^Ext(Ö/Z, G) (see §54 property H)).

In this section we shall investigate what groups G can be represented in the form G = Next(B, A) (for suitable groups A and B). Theorem 6.1 shows that such a group must necessarily be cotorsion.

Before establishing the main result of this section we prove two lemmas.

LEMMA 7.1 IfG is a reduced torsion free group then

Hom(e©n Z(p),G) = 0.

peP

Proof: Since

Hom(e © n Z(p), G) s Hom(Ö, G) © Hom(n Z(p), G),

peP peP

the lemma will follow if we can prove that both Hom(g, G) and H o m ( n Z(p),G)

peP

vanish. That H o m ( ö , G) is zero follows from the fact that G is reduced. To prove that

H o m ( n Z(p),G)

peP

is zero, consider the exact sequence

0 ^ © z(p)^ n z(p)^ n z(p)/© z(p)-o.

peP peP peP peP

From [10] we know that

®Z(p)

coincides with the maximal torsion subgroup of

nz(p)

peP

and that the factor group

nz(p)/© z(p)

peP peP

is divisible (theorem 9.2 and exercise 9.14 respectively). It follows that the sequence

Hom(n Z(p)/® Z(p), G ) ^ H o m ( n Z(p),G)^

- » H o m ( © Z(p),G)

peP

is exact. Since G is torsion free the last group in this sequence is zero, and the fact that G is reduced implies that the first group also vanishes. Hence

Hom(nz(p),G) = 0.

peP

The lemma is proved.

LEMMA 7.2 IfG is a cotorsion group then

Next(e© n2(p),G) = 0.

peP

Proof:

Next(Ö © n Z(P), G) s Next(e, G) © Next (Y\ Z(p), G),

peP peP

and since G is cotorsion Next((2, G)=0. It remains to prove that

Next(nz(p),G)

peP

is zero. Since

© Z(p)

peP

is the torsion subgroup of

Uz(p)

peP

we have from lemma 6.6 that

Next( n Z(p), G) s Next( © Z(p), G)

peP peP

S n N e x t ( Z ( p ) , G ) .

Theorem 3.3 implies that each Next(Z(p), G)=0, hence

Next(nz(p),G) = 0.

peP

This completes the proof of the lemma.

THEOREM 7.3 Let G be a reduced torsion free cotorsion group. Then there

exist groups A and B such that G is isomorphic to the group of neat ex-tensions of A by B.

Proof: Imbed Z in the divisible group Q: 0-yZ-*Q. Define a

mono-morphism a of Z into

Q®Ylizipz)

peP

by

a: z-»-(az, (..., z+pZ, ...)), zeZ.

Then a Z is a neat subgroup of

Q®lliZlpZ),

peP

for if q is any prime number and

(z,(..., z^ + pZ, ...))eQ® Y{(Z/pZ)

peP

such that

q(z, (..., Zp+pZ,...)) = az = (az, (..., z+pZ,...))

then qz = az and ^(..., z ^ + p Z , . . . ) = (..., z-l-pZ,...).

From the second equality it follows that zeqZ, say z = qz', z'eZ. Hence

az = (az, (..., z4-pZ,...)) = (a^z', (..., qz' +pZ...))

= q(az',(...,z'+pZ,...)) = q(az')eq(aZ),

proving that a Z is a neat subgroup of

Q®UiZlpZ) =

Q®UZip)-peP Q®UZip)-peP

We have proved that the sequence

O^Zxi^e© n Z(p)^(Q® n Z(p))/aZ^0

peP peP

is neat exact. Hence we have the exactness of the sequence

Hom(Q © n Z(p), G) ^ Hom(Z, G)^G-*

->Next((e© n Z(p))/aZ, G)^Next(e© H Z(p), G).

peP peP