By de Cornulier Y.

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Hence the commutator gj gi lies in G j for all integers i and j such that 1 i < j m. 3) p−1 . 3) for 1 i < j m form a presentation for G. 1) can be brought into normal form using the relations given. This can be done by one of the collection processes used in computational p-group theory. See Sims [87] for a detailed discussion of such processes. One method goes roughly as follows. 3) with i = 1. 2) with i = 1. This process results in a word of the form g1 1 g, where gm . The process is then repeated with g and the g is a word in g2 g3 generator g2 .

Thus we have shown that property (ii) follows. Finally, suppose that L is a -subgroup of G. Then LM/M is a -subgroup of G/M and so is contained in a Hall -subgroup H/M of G/M. But then H is a Hall -subgroup of G, and L H. Thus (iii) follows, and we have established the inductive step in this case. The second case is when p does not belong to . Then K/M and M are coprime. The Schur–Zassenhaus theorem implies that there is a complement H for M in K and any two complements are conjugate in K. ) But then H = K/M and so H is a Hall -subgroup of G, and so property (i) follows.

It follows that every maximal subgroup contains Gp G , and so Gp G G and G/ G is an elementary abelian group. Since G is generated by d elements, so is G/ G and so G/ G pd . If d G/ G < p then G/ G may be generated by d − 1 elements, and so G may be generated by G together with the inverse image of these elements under the natural map from G to G/ G . 9, this implies that G may be generated by d − 1 elements. This contradiction shows that G/ G = pd . We have seen that G Gp G . Since G/ Gp G is elementary abelian, the intersection of all maximal subgroups of G/ Gp G is trivial.