Test paper 4
Time allowed : 3 hours
(Allocate 20 marks for each question)
If G is a finite group and £T, K are subgroups of G prove that
\H\\K\
=
jSTTp
If G is a group of order 48 with more than one Sylow 2-subgroup,
find the possible number of Sylow 2-subgroups. If Pi, P 2 are distinct
Sylow 2-subgroups, prove that |Pi n P 2 | = 8. Show also that P1P2 C
NG{P\ nP 2 ). By considering \NG{PI ^ 2 ) ! show that Pi n P 2 is a normal
subgroup of G.
Hence show that any group of order 48 has a proper non-trivial normal
subgroup.
Show that the number of elements in a conjugacy class in a finite p-
group is a power of p. Deduce that a non-trivial finite p-group has a
non-trivial centre.
Show that if P is a non-trivial finite p-group then P contains sub-
groups P i , . . . , Pfc such that
P = Pi DP2D • • O f t = {l},
each Pi is a normal subgroup of P, and |P,- : P,-+i | = p for % = 1 , . . . , A—1.
Let G be a group with G* < Z{G). Prove that, for all x,yeG and all
integers n > 1,
Time allowed : 3 hours
(Allocate 20 marks for each question)
If G is a finite group and £T, K are subgroups of G prove that
\H\\K\
=
jSTTp
If G is a group of order 48 with more than one Sylow 2-subgroup,
find the possible number of Sylow 2-subgroups. If Pi, P 2 are distinct
Sylow 2-subgroups, prove that |Pi n P 2 | = 8. Show also that P1P2 C
NG{P\ nP 2 ). By considering \NG{PI ^ 2 ) ! show that Pi n P 2 is a normal
subgroup of G.
Hence show that any group of order 48 has a proper non-trivial normal
subgroup.
Show that the number of elements in a conjugacy class in a finite p-
group is a power of p. Deduce that a non-trivial finite p-group has a
non-trivial centre.
Show that if P is a non-trivial finite p-group then P contains sub-
groups P i , . . . , Pfc such that
P = Pi DP2D • • O f t = {l},
each Pi is a normal subgroup of P, and |P,- : P,-+i | = p for % = 1 , . . . , A—1.
Let G be a group with G* < Z{G). Prove that, for all x,yeG and all
integers n > 1,
