On groups that differ in one of four squares. (Q1864604)

Let \(\circ\) and \(*\) be two group operations on a finite set \(G\) of order \(n\). Then \(d(\circ,*)\) is defined to be the number of pairs \((u,v)\in G\times G\) with \(u\circ v\neq u*v\). It was proved by the author [Eur. J. Comb. 13, No. 5, 335-343 (1992; Zbl 0790.20038); ibid. 21, No. 3, 301-321 (2000; Zbl 0946.20006)] that \(G(\circ)\cong G(*)\) if \(d(\circ,*)\leq n^2/9\), and that, if \(n\) is a power of 2, then \(d(\circ,*)<n^2/4\) suffices. This paper is concerned with the situation where \((G,\circ)\) has subgroups \(S\) and \(H\), with \(S\) of index 2 in \(H\), which determine the places in which the Cayley tables of \((G,\circ)\) and \((G,*)\) differ, in the following sense: let \(L^\circ\) and \(R^\circ\) denote left and right cosets, then for each \((\alpha,\beta)\in L^\circ(H)\times R^\circ(H)\) there exists \((\alpha_0,\beta_0)\in L^\circ(S)\times R^\circ(S)\) with \(\alpha_0\) and \(\beta_0\) subsets of \(\alpha\) and \(\beta\) respectively, such that \(x\circ y\neq x*y\) for \((x,y)\in\alpha\times\beta\) if and only if \((x,y)\in\alpha_0\times\beta_0\). It is shown that \(S\) can always be chosen so that \(S\) is normal in both \((G,\circ)\) and \((G,*)\), and the factor groups are equal and either cyclic or dihedral.