4: Presentations
Given an abelian group G with a presentation
( xi,..., xn | ri = 1, r 2 = 1, . . . , r m = 1)
the relation matrix of the presentation is the m x n matrix A — [a,y]
where a,-y is the exponent sum of Xj in the relation ry = 1. Now A can
be reduced by elementary row and column operations, in which only
integer multiples are used, to a diagonal matrix D = diag{di,... ,dt}
where t = min{n, m}. This is equivalent to finding invertible integer
matrices P,Q such that PAQ~l = D. We can assume that di,...,ck
are non-zero and ck+i,..., dt are zero. Then if C is the direct product
of n — k copies of C^ we have
G~CdlxCd2x --xCdkx C.
If G is a group then G/G is abelian, and G is called perfect if GjG1 is
1
the trivial group. If G is given by the presentation (X \ R) then G/G'
is given by the presentation (X | R, C) where
C={[xi,xj] | Xi.XjGX}.
This is a special case of von Dyck's theorem which shows that adding
relations to a group presentation leads to a quotient group. In fact,
von Dyck's theorem lies behind the method of showing that a given
presentation defines a particular group.
4.1 Let G be the abelian group
(a,6,c | a 37 6 27 c 47 = a52fc37c67 = a59fe44c74 - 1,
ab = 6a, be — c6, ca = ac).
, 4: Presentations
Express G as a direct product of cyclic groups.
By adding the relation a 3 6 2 c 4 = 1 to those of G show that a 3 6 2 c 4 is
not the identity of G. Deduce that a3b2c4 is an element of order 5 in G.
Find an element of order 7 and an element of order 35 in G.
4.2 Express each of the following as a direct product of cyclic groups :
(a) (a,6,c | a 2 6 3 c 6 = a 4 6 9 c 4 = 1, ab — 6a, be = cb, ca = ac);
(b) (a,6,c | a 2 6 3 c 6 = a 4 fcV = a3b3c2 = 1, at = 6a, 6c = c6, ac =
ca).
4.3 Let G be the group with presentation
G= (a, 6 | a6a62 = 1).
Show that G is abelian. Describe the structure of G.
4.4 Let G be an abelian group. Show that the elements of finite order form
a subgroup T. Let Q consist of the elements of infinite order together
with the identity element. Find a necessary and sufficient condition for
Q to be a subgroup of G.
Let IT be the set of primes and define a group G as follows. Let
X — Upen ^-P a n d kt G be the set of mappings / : II —> X such that
f(p) G Z p for every p G II. Given /, g G G define / -f g : II —• X to be
the mapping given by
Show that G is an abelian group containing elements of prime order for
every prime, and elements of infinite order.
Prove that the subgroup T in this case consists of those mappings
/ G G with the property that there are only finitely many p G II with
f(p) * o.
4.5 Let G be the group with presentation
G = ( a , 6 , c | anbmcm = ambncm = ambmcn = 1).
Prove that G/G' is infinite if and only if m = n or 2m = — ft.
Show further that (3 is perfect if and only if G is the trivial group.
4.6 If n is an integer that is coprime to 6 show that there is an integer k
such that the group
(x,y | x2 = {xyf, (sy'syi^
is perfect.
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Given an abelian group G with a presentation
( xi,..., xn | ri = 1, r 2 = 1, . . . , r m = 1)
the relation matrix of the presentation is the m x n matrix A — [a,y]
where a,-y is the exponent sum of Xj in the relation ry = 1. Now A can
be reduced by elementary row and column operations, in which only
integer multiples are used, to a diagonal matrix D = diag{di,... ,dt}
where t = min{n, m}. This is equivalent to finding invertible integer
matrices P,Q such that PAQ~l = D. We can assume that di,...,ck
are non-zero and ck+i,..., dt are zero. Then if C is the direct product
of n — k copies of C^ we have
G~CdlxCd2x --xCdkx C.
If G is a group then G/G is abelian, and G is called perfect if GjG1 is
1
the trivial group. If G is given by the presentation (X \ R) then G/G'
is given by the presentation (X | R, C) where
C={[xi,xj] | Xi.XjGX}.
This is a special case of von Dyck's theorem which shows that adding
relations to a group presentation leads to a quotient group. In fact,
von Dyck's theorem lies behind the method of showing that a given
presentation defines a particular group.
4.1 Let G be the abelian group
(a,6,c | a 37 6 27 c 47 = a52fc37c67 = a59fe44c74 - 1,
ab = 6a, be — c6, ca = ac).
, 4: Presentations
Express G as a direct product of cyclic groups.
By adding the relation a 3 6 2 c 4 = 1 to those of G show that a 3 6 2 c 4 is
not the identity of G. Deduce that a3b2c4 is an element of order 5 in G.
Find an element of order 7 and an element of order 35 in G.
4.2 Express each of the following as a direct product of cyclic groups :
(a) (a,6,c | a 2 6 3 c 6 = a 4 6 9 c 4 = 1, ab — 6a, be = cb, ca = ac);
(b) (a,6,c | a 2 6 3 c 6 = a 4 fcV = a3b3c2 = 1, at = 6a, 6c = c6, ac =
ca).
4.3 Let G be the group with presentation
G= (a, 6 | a6a62 = 1).
Show that G is abelian. Describe the structure of G.
4.4 Let G be an abelian group. Show that the elements of finite order form
a subgroup T. Let Q consist of the elements of infinite order together
with the identity element. Find a necessary and sufficient condition for
Q to be a subgroup of G.
Let IT be the set of primes and define a group G as follows. Let
X — Upen ^-P a n d kt G be the set of mappings / : II —> X such that
f(p) G Z p for every p G II. Given /, g G G define / -f g : II —• X to be
the mapping given by
Show that G is an abelian group containing elements of prime order for
every prime, and elements of infinite order.
Prove that the subgroup T in this case consists of those mappings
/ G G with the property that there are only finitely many p G II with
f(p) * o.
4.5 Let G be the group with presentation
G = ( a , 6 , c | anbmcm = ambncm = ambmcn = 1).
Prove that G/G' is infinite if and only if m = n or 2m = — ft.
Show further that (3 is perfect if and only if G is the trivial group.
4.6 If n is an integer that is coprime to 6 show that there is an integer k
such that the group
(x,y | x2 = {xyf, (sy'syi^
is perfect.
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