Test paper 3
Time allowed : 3 hours
(Allocate 20 marks per question)
Let G be a finite group and let p be a prime dividing the order of G.
Let P%,..., P r be the Sylow p-subgroups of G. Show that the mapping
fig from { P i , . . . , Pr} to itself defined by
is a bijection. Show also that the mapping 0 from G to the group of bi-
jections on { P i , . . . , P r } given by ®(g) = fig is a morphism whose kernel
is the largest normal subgroup of G that is contained in the normaliser
in G of a Sylow p-subgroup.
Let ( ? b e a group of order 168 which has no non-trivial proper nor-
mal subgroups. Show that G cannot be represented non-trivially as a
permutation group on fewer than seven letters. Show that G can be
represented as a permutation group on eight letters.
Let G be a nilpotent group and H a normal abelian subgroup of G with
the property that H is not properly contained in any normal abelian
subgroup of G. Prove that H={geG \ (\/h e H) [gt h] = 1}.
Deduce that H is not properly contained in any abelian subgroup of
G and that Aut G contains a subgroup isomorphic to G/H.
Show that if p is prime then t| : 2 —» 2 / p 2 induces a morphism from
G* = SL(2,2) to the group G* = SL(2,2 P ).
Given that in both G* and G* the centre is the subgroup generated
by
r
-l o
0 -1
Time allowed : 3 hours
(Allocate 20 marks per question)
Let G be a finite group and let p be a prime dividing the order of G.
Let P%,..., P r be the Sylow p-subgroups of G. Show that the mapping
fig from { P i , . . . , Pr} to itself defined by
is a bijection. Show also that the mapping 0 from G to the group of bi-
jections on { P i , . . . , P r } given by ®(g) = fig is a morphism whose kernel
is the largest normal subgroup of G that is contained in the normaliser
in G of a Sylow p-subgroup.
Let ( ? b e a group of order 168 which has no non-trivial proper nor-
mal subgroups. Show that G cannot be represented non-trivially as a
permutation group on fewer than seven letters. Show that G can be
represented as a permutation group on eight letters.
Let G be a nilpotent group and H a normal abelian subgroup of G with
the property that H is not properly contained in any normal abelian
subgroup of G. Prove that H={geG \ (\/h e H) [gt h] = 1}.
Deduce that H is not properly contained in any abelian subgroup of
G and that Aut G contains a subgroup isomorphic to G/H.
Show that if p is prime then t| : 2 —» 2 / p 2 induces a morphism from
G* = SL(2,2) to the group G* = SL(2,2 P ).
Given that in both G* and G* the centre is the subgroup generated
by
r
-l o
0 -1
