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Published online by Cambridge University Press: 05 August 2013
Abstract
If G(*) and G(∘) are two different groups on the same set, then their distance is defined as the number of pairs (a, b) ∈ G × G with a * b ≠ a ∘ b. Questions about minimal possible distances lead to different problems as isomorphic groups are or are not admitted. The paper presents many facts known about these problems and establishes the minimal distance of an element ary-ab elian 2-group from non-isomorphic groups of the same order.
Introduction
The purpose of this paper is to report on facts known about (Hamming) distances of finite groups and to prove some new results in this area.
If G(∘) and G(*) are two groups on G, then their distance dist(G(∘), G(*)) is defined to be the number of pairs (a, b) ∈ G × G with a ∘ b ≠ a * b. For a fixed group G(∘) define δ(G(∘)) to be the minimum of dist(G(∘), G(*)), where G(*) runs through all groups on G that are different from G(∘). Similarly, if G(*) runs through all groups on G that are not isomorphic to G(∘), then the minimum of dist(G(∘), G(*)) will be denoted by v(G(∘)).
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