1: Subgroups
The isomorphism and correspondence theorems for groups should be
familiar to the reader. The first isomorphism theorem (that if / : G —• H
is a group morphism then G/ Ker/ ~ I m / ) is a fundamental result from
which follow further isomorphisms : if A < G (i.e. A is a subgroup of
G), if TV" < G (i.e. N is a normal subgroup of G), and if K < G with
K < N, then
A/(A n N) ~ NA/N and G/N ~ {G/K)/{N/K).
The correspondence theorem relates the subgroups of G/N to the sub-
groups of G that contain N.
Elements a, 6 of G are said to be conjugate if a = g~lbg for some
g G G. Conjugacy is an equivalence relation on G and the correspond-
ing classes are called conjugacy classes. The subset of G consisting of
those elements that belong to singleton conjugacy classes forms a normal
subgroup Z(G) called the centre of G. For H < G the subset
MG{H) = {geG | {VheH) g-lhg e H)
is called the normaliser of H in G. It is the largest subgroup of G in which
H is normal. The derived group of G is the subgroup G1 generated by
all the commutators [a, 6] = a~1b~1ab in G, and is the smallest normal
subgroup of G with abelian quotient group.
Examples are most commonly constructed with groups of matrices
(subgroups of the group GL(n, F) of invertible nxn matrices with entries
in a field F), groups of permutations (subgroups of the symmetric groups
Sn)j groups given by generators and relations, and direct (cartesian)
products of given groups.
, Book 5 Groups
An example of a presentation is
G={a,b | a2 = 63 = 1, a~Ha = 6 " 1 ) .
Since |{6)| = 3 and (6) < G with G/(b) ~ C^ (the cyclic group of order
2), we see that \G\ = 6 . The generators a and b can be taken to corre-
spond to the permutations (1 2) and (12 3) which generate S$, or to the
matrices
r i r
1 1
1 0
which generate SL(2,7L<2,), the group of 2 x 2 matrices of determinant 1
with entries in the field Z 2 - Thus we have that G ~ S3 ~ SL(2, Z 2 ).
1.1 Let G be a group, let H be a subgroup of G and let if be a subgroup
of H. Prove that
Deduce that the intersection of a finite number of subgroups of finite
index is a subgroup of finite index. Is the intersection of an infinite
number of subgroups of finite index necessarily also of finite index?
1.2 Let G be a group and let H be a subgroup of G. Prove that the only
left coset of H in G that is a subgroup of G is H itself. Prove that the
assignment
ip:xH^ Ex~l
describes a mapping from the set of left cosets of H in G to the set of
right cosets of H. Show also that ip is a bijection. Does the prescription
ij) : xH f-> Hx
describe a mapping from the set of left cosets of H to the set of right
cosets of HI If so, is ip a bijection?
1.3 Find a group G with subgroups H and K such that HK is not a sub-
group.
1.4 Consider the subgroup H = ((1 2)) of £ 3 . Show how the left cosets of H
partition 53. Show also how the right cosets of H partition S3. Deduce
that H is not a normal subgroup of 53.
1.5 Let G be a group and let H be a subgroup of G. If g £ G is such that
\(g)\ = n and gm G H where m and n are coprime, show that g E H.
The isomorphism and correspondence theorems for groups should be
familiar to the reader. The first isomorphism theorem (that if / : G —• H
is a group morphism then G/ Ker/ ~ I m / ) is a fundamental result from
which follow further isomorphisms : if A < G (i.e. A is a subgroup of
G), if TV" < G (i.e. N is a normal subgroup of G), and if K < G with
K < N, then
A/(A n N) ~ NA/N and G/N ~ {G/K)/{N/K).
The correspondence theorem relates the subgroups of G/N to the sub-
groups of G that contain N.
Elements a, 6 of G are said to be conjugate if a = g~lbg for some
g G G. Conjugacy is an equivalence relation on G and the correspond-
ing classes are called conjugacy classes. The subset of G consisting of
those elements that belong to singleton conjugacy classes forms a normal
subgroup Z(G) called the centre of G. For H < G the subset
MG{H) = {geG | {VheH) g-lhg e H)
is called the normaliser of H in G. It is the largest subgroup of G in which
H is normal. The derived group of G is the subgroup G1 generated by
all the commutators [a, 6] = a~1b~1ab in G, and is the smallest normal
subgroup of G with abelian quotient group.
Examples are most commonly constructed with groups of matrices
(subgroups of the group GL(n, F) of invertible nxn matrices with entries
in a field F), groups of permutations (subgroups of the symmetric groups
Sn)j groups given by generators and relations, and direct (cartesian)
products of given groups.
, Book 5 Groups
An example of a presentation is
G={a,b | a2 = 63 = 1, a~Ha = 6 " 1 ) .
Since |{6)| = 3 and (6) < G with G/(b) ~ C^ (the cyclic group of order
2), we see that \G\ = 6 . The generators a and b can be taken to corre-
spond to the permutations (1 2) and (12 3) which generate S$, or to the
matrices
r i r
1 1
1 0
which generate SL(2,7L<2,), the group of 2 x 2 matrices of determinant 1
with entries in the field Z 2 - Thus we have that G ~ S3 ~ SL(2, Z 2 ).
1.1 Let G be a group, let H be a subgroup of G and let if be a subgroup
of H. Prove that
Deduce that the intersection of a finite number of subgroups of finite
index is a subgroup of finite index. Is the intersection of an infinite
number of subgroups of finite index necessarily also of finite index?
1.2 Let G be a group and let H be a subgroup of G. Prove that the only
left coset of H in G that is a subgroup of G is H itself. Prove that the
assignment
ip:xH^ Ex~l
describes a mapping from the set of left cosets of H in G to the set of
right cosets of H. Show also that ip is a bijection. Does the prescription
ij) : xH f-> Hx
describe a mapping from the set of left cosets of H to the set of right
cosets of HI If so, is ip a bijection?
1.3 Find a group G with subgroups H and K such that HK is not a sub-
group.
1.4 Consider the subgroup H = ((1 2)) of £ 3 . Show how the left cosets of H
partition 53. Show also how the right cosets of H partition S3. Deduce
that H is not a normal subgroup of 53.
1.5 Let G be a group and let H be a subgroup of G. If g £ G is such that
\(g)\ = n and gm G H where m and n are coprime, show that g E H.
