 By Ji L.

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Extra resources for 2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four

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Let P be a Sylow subgroup of G, and assume, seeking a contradiction, that NG P is a proper subgroup of G. Then M for some maximal subgroup M of G. Now P M, and so P is NG P a Sylow subgroup of the normal subgroup M. 7]), NG P M = G. But NG P M and so we have our required contradiction. Thus every Sylow subgroup is normal in G and so G is nilpotent. 4 Linear algebra This section contains some enumeration results from linear algebra, rehearses some of the standard material on alternating forms and draws out some of the 9780521882170c03 July-2007 20 Page-20 Preliminaries immediate consequences for elementary abelian p-groups.

1 Hall subgroups and Sylow systems Let G be a finite group. A subgroup H of G is said to be a Hall subgroup if H and G H are coprime. Let be a set of prime numbers. A subgroup H of G is a Hall -subgroup if H is a number (a product of primes in number (a product of primes not in ). 1 Let G be a soluble group of order p1 1 · · · pk k . Let r be an pr . Then: integer such that 1 r k and let = p1 p2 (i) The group G has a Hall -subgroup (of order p1 1 · · · pr r ). (ii) Any two Hall -subgroups are conjugate in G.

For each choice of i j k u v, we know that 0 c i j k u v pa u v − 1. 4, and so we may choose the integers c i j k 2 v subject to the additional condition that the minimum dimension of a subspace U of V such that U U = W is s. 5, there are pNp r1 r2 s choices for a map V × V → W such that the minimum dimension of a subspace U of V satisfying U U = V is s. 12) j=1 k=j+1 v=1 since there are r21 choices for j and k. 9) with i = 1. We now give an upper bound on the number of choices for the integers c i j k u v when i > 1.