Near 2-groups yield an isomorphism with many fixed points
type: Item
instance of: scholarly article
title: Near 2-groups yield an isomorphism with many fixed points
zbMATH Open document ID: 1025.20010
DOI: 10.1016/S0012-365X(02)00809-9
author: Aleš Drápal
published in: Discrete Mathematics
publication date: 2003-06-09T00:00:00Z
review text: 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.
reviewed by: Sheila Oates-Williams
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 1924767
zbMATH Keywords: group multiplication tables
zbMATH Keywords: 2-groups
zbMATH Keywords: Hamming distance
zbMATH Keywords: distances of finite groups
MaRDI profile type: Publication
cites work: Q4221869
cites work: How far apart can the group multiplication tables be?
cites work: Q4240521
cites work: Non-isomorphic \(2\)-groups coincide at most in three quarters of their multiplication tables
cites work: Q4263751
full work available at URL: s0012-365x(02)00809-9
OpenAlex ID: W2085039756
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description: scientific article; zbMATH DE number 1924767