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Published online by Cambridge University Press: 11 January 2010
Abstract
This paper is concerned with finite groups G(∘) and G(*) of order n that are not isomorphic, and where the size of {(u, v) ∈ G×G; u∘v ≠ u*v} is the least possible (with respect to the given n). It surveys the case of 2-groups, discusses the possible generalization of the known results to p-groups, p an odd prime, and establishes the least possible distance in the case when G(*) is an elementary abelian 3-group.
Let G(∘) and G(*) be finite groups of order n. Consider the set {(u, v) ∈ G×G; u∘v ≠ u*v} and denote its size by d(∘, *). The number d(∘, *) is called the (Hamming) distance of ∘ and *. If d(∘, *) < n2/4 and n is a power of two, then G(∘) ≅ G(*), by [4]. Section 1 lists further results about distances of 2-groups, while Section 2 discusses the associated proof machinery, and its possible generalization to p-groups, p an odd prime. In Section 2 there are also presented non-isomorphic p-groups G(∘) and G(*), |G| = n > p, for which d(∘, *) = n2(p2 − 1)/(4p2). This result is the best possible, when G(∘) is an elementary abelian 3-group—in Section 3 we shall show that in such a case d(∘, *) < 2n2/9 implies G(∘) ≅ G(*).
If H ≤ G(∘), then the set of all left (or right) cosets of H in G(∘) is denoted by L∘(H) and R∘(H), respectively.
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