28 1. Sylow Theory

ID . 11. Let n be the maximum of the orders of the abelian subgroups of a

finite group G. Show that \G\ divides n!.

Hint. Show that for each prime p, the order of a Sylow p-subgroup of G

divides n\.

Note. There exist infinite groups in which the abelian subgroups have

bounded order, so finiteness is essential here.

ID.12. Let p be a prime dividing the order of a group G. Show that the

number of elements of order p in G is congruent to —1 modulo p.

ID.13. If Z C Z(G) and G/Z is nilpotent, show that G is nilpotent.

ID . 14. Show that the Frattini subgroup &(G) of a finite group G is nilpo-

tent.

Hint. Apply the Frattini argument. The proof here is somewhat similar to

the proof that (3) implies (4) in Theorem 1.26.

Note. This problem shows that $(G) CF(G).

ID.15. Suppose that $(G) C N G and that N/$(G) is nilpotent. Show

that N is nilpotent. In particular, if G/${G) is nilpotent, then G is nilpo-

tent.

Note. This generalizes the previous problem, ^yhich follows by setting N =

$(G). Note that this problem proves that F(G/$(G)) = F(G)/$(G).

1D.16. Let N G, where G is finite. Show that $(JV) C J(G).

Hint. If some maximal subgroup M of G fails to contain 4(7V), then

$(N)M = G, and it follows that N = $(N)(N n M).

ID.17. Let N G, where N is nilpotent and G/N' is nilpotent. Prove that

G is nilpotent.

Hint. The derived subgroup

TV7

is contained in fr(7V) by Problem ID.8.

Note. If we weakened the assumption that G/N' is nilpotent and assumed

instead that G/N is nilpotent, it would not follow that G is necessarily

nilpotent.

1D.18. Show that F(G/Z(G)) = F(G)/Z(G) for all finite groups G.

ID.19. Let F — F(G), where G is an arbitrary finite group, and let C =

CG(F).

Show that C/(C fl F) has no nontrivial abelian normal subgroup.

Hint. Observe that F(G) G.