Relative relation modules of finite groups

BULL. AUSTRAL. MATH. SOC. VOL.

20C05,

2OFO5

28 ( 1 9 8 3 ) , 4 4 7 - 4 4 8 .

RELATIVE RELATION MODULES OF FINITE GROUPS M. YAMIN

Let

G be a fixed finite group and consider a short exact sequence G

where

E

is a finitely generated group.

•* 1

The abelian group

be regarded as a7L -module and, for a fixed prime u abelian

p-group

S = S/S 'Sr = S/pS

E is a free group, if) , and

S

S

the relation module modulo 5

p .

In general we call

G and

If

G determined by S

the

the relative relation module modulo

When the minimal number of generators of S

may

may be regarded as an IF G-module.

i s called the relation module of

relative relation module, and

S = S/S'

p , the elementary

E is the same,

S

p . and

will be called minimal. Gaschiitz, Gruenberg and others have studied relation modules and

relation modules modulo

p .

The main aim of this thesis is to study

relative relation modules modulo

p

groups.

To be more precise, l e t

X = {g., 1 < i 5 d}

of

G.

the

G, G. ,

when

the cyclic group generated by

1 5 i 5 d ,

£

is a free product of cyclic

g. ,

be a generating set

E the free product of

i|» the epimorphism whose restriction to each

is the identity isomorphism, and

S

G.

the kernel of ty .

Some of the results may be summarised as follows.

S

i s embedded in

Received 25 October 1983. Thesis submitted to Australian National University, March 1983. Degree approved, August 1983- Supervisor: Dr P.J. Cossey. Copyright Clearance Centre, Inc. Serial-fee code: $A2.00 + 0.00.

000U-972T/83

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448

M. Yamin

the direct sum of the augmentation ideals of the

IF G. , 1 « i 5 d , P "l G , and the resulting factor module is isomorphic to the

induced to

augmentation ideal of of rank

IF G . P

S

may also be embedded in a free

IF G-module P

d - 1 .

Two relative relation modules, isomorphic as isomorphic as but also on

C-modules; \\i .

that i s ,

Some cases when

5 S

IF -spaces, are rarely

not only depends on does not depend on

G, p

and d

\p are

established. We say that order of

p

is semicoprime to the order of

G if

G and does not divide the orders of the

G. ,

p

divides the

1 5 i 5 d .

In

"If

the coprime and semicoprime cases a characterisation (including a criterion for counting protective summands) of modules (modulo completely; groups of

p ) of

SL(2, p)

S

and

is given. PSL(2, p)

Some relative relation are described

the description may be useful in the study of the factor PSL(2, 2Z) .

Given an unrefinable direct decomposition of a module, the direct sum of a l l the nonprojective summands is called the nonprojective part of the module.

In the semicoprime case the nonprojective part of

5

is a

uniquely determined, nonzero and indecomposable module (and is also the nonprojective part of part of

S

S

when

E is a free group).

The nonprojective

in the nons end coprime case may be zero or decomposable (and may

not be a homomorphic image of the nonprojective part of free).

When

G is a

indecomposable for

5

p-group, we prove that

5

S

when

E

is

is nonprojective and

minimal.

Some of the above results can tie generalised in the case when the cyclic factors of

E are not restricted to be the generators of

G ,

however i t is not known whether in this case the minimal relative relation modules of

p-groups are also indecomposable.

Some results may also be extended to

S .

Department of Mathematics, Centre for Advanced Study in Mathematics, Panjab University, Chandigarh 160014, India. Downloaded from https://www.cambridge.org/core. IP address: 207.90.37.11, on 06 Nov 2017 at 17:56:06, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700021183