On Distances of 2-groups and 3-groups

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 6= 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 6= 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 2groups, 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. If A ⊆ G and B ⊆ G, then the size of {(u, v) ∈ A×B; u ◦ v 6= u ∗ v} will be denoted by d(A,B).