Near 2-groups yield an isomorphism with many fixed points
Let \(\circ\) and \(*\) be two group operations on a finite set \(G\). Then the `Hamming distance', \(d(\circ,*)\), of \(\circ\) and \(*\) is the number of pairs \((u,v)\in G\times G\) with \(u\circ v\neq u*v\). It was proved by this author [Eur. J. Comb. 13, No. 5, 335-343 (1992; Zbl 0790.20038)] that \(G(\circ)\cong G(*)\) if \(d(\circ,*)\leq n^2/9\), and that in such a case there is an isomorphism which fixes more than \(15n/16\) elements of \(G\). The author also proved [ibid. 21, No. 3, 301-321 (2000; Zbl 0946.20006)] that in the case of 2-groups, \(d(\circ,*)<n^2/4\) suffices to force isomorphism, but gave no estimate of the number of fixed elements. This lacuna is addressed in this paper, the main theorem of which states that in such a situation there is an isomorphism which fixes more than \((3+1/\sqrt 3)n/4\) elements of \(G\). In the case of groups of order 16, the above condition implies that any isomorphism fixes at least 15 elements, and so is the identity, so two different multiplication tables of groups of order 16 must differ in at least 64 places.
scientific article; zbMATH DE number 1924767