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.
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