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On Groups that Differ in One of Four Squares

✍ Scribed by Aleš Drápal

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
371 KB
Volume
23
Category
Article
ISSN
0195-6698

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✦ Synopsis

For a subgroup T of a group G(•), let L • (T ) and R • (T ) denote the sets of all left and right cosets, respectively. This paper is concerned with finite groups G(•) and G( * ), where the places in which the Cayley tables of the two groups differ is determined by subgroups

so that α 0 ⊆ α and β 0 ⊆ β, and so that x • y = x * y holds for (x, y) ∈ α × β if and only if (x, y) ∈ α 0 × β 0 . Given G(•) and G( * ), there can be multiple choices of S and H and it is proved in the paper that there always exists a choice for which S is a normal subgroup of both G(•) and G( * ), and G(•)/S = G( * )/S is either cyclic or dihedral (where the latter includes Klein's four-element group). The specification of S and H is precise enough to permit a detailed description of the set of products for which • and * differ and of the way in which they differ and, moreover, to permit the derivation of G( * ) from G(•) (without knowing G( * ) in advance).

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