By Paley R.E.A.C., Wiener N.
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Let x ∈ NG H be such that xH has order p in NG H /H. The last assertion of the lemma now follows by taking K to be the group generated by H and x. 11 Let G be a finite p-group. Then every maximal subgroup M of G is normal and has index p in G. 12 Let G be a finite p-group. Then G/ G is an elementary abelian p-group of order pd , where d is the minimal number of generators of G. Moreover G = Gp G , where Gp is the subgroup generated by the p set x x ∈ G . 11, every maximal subgroup M of G is normal and has index p in G.
This enables us to show that a large number of isomorphism classes of groups occur as quotients of the group G, and this will provide the lower bound we need. 1 Relatively free groups Let r be a positive integer. Let Fr be a free group of rank r, generated by x1 x2 xr . Let Gr be the quotient of Fr by the subgroup N generated by 2 all words of the form xp , x y p and x y z . 3. The group Gr is the relatively free group in the variety of p-groups of -class 2; see Hanna Neumann  for an introduction to varieties of groups in general.
Proof: Since any commutator or pth power in Gr is central of order p, we have that Gpr Gr is a central elementary abelian p-group. Since Gr /Gpr Gr is also an elementary abelian p-group, Gr is a p-group. 12, Gr = Gpr Gr , and so Gr is central. It is not difficult to show that Gr is generated by the elements xip for i ∈ 1 2 r and xj xi where 1 i < j r. These elements form a minimal generating set of p−1 Gr ; we can see this as follows. 1) 1 i