Test paper 2
Time allowed : 3 hours
(Allocate 20 marks for each question)
1 Let ( j b e a finite group of order pmn where p is a prime that is cop rime
to n. What do the three Sylow theorems tell you about the p-subgroups
of G?
Show, by using induction on the order or otherwise, that a maximal
subgroup of a finite p-group P is normal in P .
Supose that G has at least three Sylow p-subgroups Pi,P2,P3 where
Pi n P 2 and P2C\Pz are maximal subgroups of index p in P 2 . Show that
Pi = (hk)-lPzhk where h G MG{P2 n P 3 ) and k G MG{Pi n P 2 ).
2 Prove that every subgroup of a nilpotent group is subnormal. Deduce
that a maximal subgroup of a nilpotent group is normal.
Let G b e a group in which every finitely generated subgroup is nilpo-
tent, and let M be a maximal subgroup of G. Suppose that M is not
normal in G. Prove that there is an element x of G' with x £ M. Writ-
ing x = Iir=i \y*i zi\> P r o v e that {xj yt, Zi | % = 1,..., n) is contained in
a subgroup H of G where
H = {x) d\,..., a m | chi G Af, i = 1 , . . . , TYI ).
Let A = {a\,..., am ) and let L be maximal in H with respect to the
property that A < L and x ^ L. Show that L is a maximal subgroup of
H, that xeH', and that H' < L.
Deduce from the above that a maximal subgroup of a group in which
every finitely generated subgroup is nilpotent is normal.
Time allowed : 3 hours
(Allocate 20 marks for each question)
1 Let ( j b e a finite group of order pmn where p is a prime that is cop rime
to n. What do the three Sylow theorems tell you about the p-subgroups
of G?
Show, by using induction on the order or otherwise, that a maximal
subgroup of a finite p-group P is normal in P .
Supose that G has at least three Sylow p-subgroups Pi,P2,P3 where
Pi n P 2 and P2C\Pz are maximal subgroups of index p in P 2 . Show that
Pi = (hk)-lPzhk where h G MG{P2 n P 3 ) and k G MG{Pi n P 2 ).
2 Prove that every subgroup of a nilpotent group is subnormal. Deduce
that a maximal subgroup of a nilpotent group is normal.
Let G b e a group in which every finitely generated subgroup is nilpo-
tent, and let M be a maximal subgroup of G. Suppose that M is not
normal in G. Prove that there is an element x of G' with x £ M. Writ-
ing x = Iir=i \y*i zi\> P r o v e that {xj yt, Zi | % = 1,..., n) is contained in
a subgroup H of G where
H = {x) d\,..., a m | chi G Af, i = 1 , . . . , TYI ).
Let A = {a\,..., am ) and let L be maximal in H with respect to the
property that A < L and x ^ L. Show that L is a maximal subgroup of
H, that xeH', and that H' < L.
Deduce from the above that a maximal subgroup of a group in which
every finitely generated subgroup is nilpotent is normal.
