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Infinite groups in a given variety and Ramsey’s theorem. (English) Zbl 0751.20020
The paper contains as its main results two group-theoretic theorems which are reminiscent of Ramsey’s Theorem in combinatorics and, in fact, use Ramsey’s Theorem in their proofs. The first result is the following. Let \(G\) be an infinite group and \(k\) a positive integer, \(k>1\). If for every sequence \(X_ 1,\dots,X_ k\) of \(k\) infinite subsets of \(G\) there exist elements \(x_ i\) of \(X_ i\), \(i=1,\dots,k\), such that \([x_ 1,\dots,x_ k]=1\) then \(G\) is nilpotent of class at most \(k-1\). (As usual \([x_ 1,\dots,x_ k]\) denotes the left-normed group commutator.) The authors also discuss an analogous general problem for varieties of groups which is solved by their result in the case of the variety of all nilpotent groups of class at most \(k-1\). The second main result of the paper concerns so-called “rewritable” groups. Let \(G\) be an infinite group and \(n\) a positive integer, \(n>1\). If for every sequence \(X_ 1,\dots,X_ n\) of \(n\) infinite subsets of \(G\) there exist elements \(x_ i\) of \(X_ i\), \(i=1,\dots,n\), and a non-trivial permutation \(\sigma\) of \(\{1,\dots,n\}\) such that \(x_ 1\dots x_ n=x_{\sigma(1)}\dots x_{\sigma(n)}\) then for every sequence \(x_ 1,\dots,x_ n\) of \(n\) elements of \(G\) there exists a non-trivial permutation \(\sigma\) of \(\{1,\dots,n\}\) such that \(x_ 1\dots x_ n=x_{\sigma(1)}\dots x_{\sigma(n)}\). It follows that \(G\) is finite-by-abelian-by-finite.

MSC:
20E10 Quasivarieties and varieties of groups
20F12 Commutator calculus
20F18 Nilpotent groups
20E34 General structure theorems for groups
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References:
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[7] DOI: 10.1017/S1446788700019303
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