Test paper 1
Time allowed : 3 hours
(Allocate 20 marks for each question)
Prove that the centre of a group of order pn is non-trivial. Let if be a
finite group and let H be a subgroup of if. If Pi is a Sylow p-subgroup
of H, explain why Pi < P 2 for some Sylow p-subgroup P 2 of K.
Now suppose that H satisfies the condition that if h £ H and h ^ 1
then MKW < H. By considering the centre of P2, or otherwise, show
that Pi = P 2 .
Deduce that h.c.f.(|#|, \K : H\) = 1.
Let the quaternion group Qs be given by the presentation
Qs = (a,6 I a4 = 1, a2 = 62, a6a = 6).
Show that the mappings a, /3 defined by
a(a) = ab, a(b) = a
/J(a) = 6, /?(&) = a
extend to automorphisms of Q8-
Let G = (a, /?}. Prove that G is a group of order 24 isomorphic to
£4. Show also that
G~
Suppose that A is a set of generators of a group G and that H is a
proper subgroup of G. Given an element a of A not belonging to JET, let
j? be the set obtained from A by replacing each x € A n if by ax. Show
Time allowed : 3 hours
(Allocate 20 marks for each question)
Prove that the centre of a group of order pn is non-trivial. Let if be a
finite group and let H be a subgroup of if. If Pi is a Sylow p-subgroup
of H, explain why Pi < P 2 for some Sylow p-subgroup P 2 of K.
Now suppose that H satisfies the condition that if h £ H and h ^ 1
then MKW < H. By considering the centre of P2, or otherwise, show
that Pi = P 2 .
Deduce that h.c.f.(|#|, \K : H\) = 1.
Let the quaternion group Qs be given by the presentation
Qs = (a,6 I a4 = 1, a2 = 62, a6a = 6).
Show that the mappings a, /3 defined by
a(a) = ab, a(b) = a
/J(a) = 6, /?(&) = a
extend to automorphisms of Q8-
Let G = (a, /?}. Prove that G is a group of order 24 isomorphic to
£4. Show also that
G~
Suppose that A is a set of generators of a group G and that H is a
proper subgroup of G. Given an element a of A not belonging to JET, let
j? be the set obtained from A by replacing each x € A n if by ax. Show
