On Groups that Differ in One of Four Squares

For a subgroup T of a group G( ∘ ), letL ∘ (T) andR ∘ (T) denote the sets of all left and right cosets, respectively. This paper is concerned with finite groups G( ∘ ) andG ( * ), where the places in which the Cayley tables of the two groups differ is determined by subgroups S < H ≤ G( ∘ ), such that |H : S | = 2, in the sense that for all (α, β) ∈ L ∘ (H) × R ∘ (H) one can find (α0,β0 ) ∈ L ∘ (S) × R ∘ (S) so thatα0 ⊆ α and β0 ⊆ β , and so that x ∘ y ≠ = x * y holds for (x, y) ∈ α × β if and only if (x, y) ∈ α0 × β0. GivenG ( ∘ ) and G( * ), there can be multiple choices ofS 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( * ), andG ( ∘ ) / 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( * ) fromG ( ∘ ) (without knowing G( * ) in advance).