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Criteria for commutativity in large groups. (English) Zbl 0908.20022

Let \(m\), \(n\) be positive integers. Then a group \(G\) is called an \((m,n)\)-group if \((XY)^n=(YX)^n\) for all subsets \(X\), \(Y\) of \(G\) with \(m\) elements. (Here \(X^n\) is the set of all \(x_1x_2\cdots x_n\), \(x_i\in X\).) It is known that a \((1,n)\)-group \(G\) is abelian or else \(G^n\leq Z(G)\).
In the present work the authors consider the case where \(m>1\). The main results are as follows. Theorem 1: Let \(m\geq 2\), \(n\geq 1\), and let \(G\) be an \((m,n)\)-group which is not abelian. Then \(G\) is a BFC-group with finite exponent bounded by a function of \(m\) and \(n\). Also, if \(m\geq n\), then \(G\) has finite order bounded by a function of \(m\) and \(n\). [A BFC-group is a group whose conjugacy classes are boundedly finite.] Theorem 2: The only non-abelian \((2,2)\)-group is the quaternion group of order \(8\).
A group \(G\) is called \((m,n)\)-permutable if \(X_1\cdots X_n\subseteq\bigcup_{\sigma\in S_n\setminus 1} X_{\sigma(1)}\cdots X_{\sigma(n)}\) for all \(m\)-element subsets \(X_i\) of \(G\). Thus the \((1,n)\)-permutable groups \(G\) are those such that \(x_1x_2\ldots x_n=x_{\sigma(1)}x_{\sigma(2)}\ldots x_{\sigma(n)}\) for all \(x_i\in G\) and some \(\sigma\neq 1\) in \(S_n\); these groups are called \(n\)-permutable and their structure is well-understood. Theorem 3: Let \(m,n\geq 1\) and let \(G\) be an \((m,n)\)-permutable group. Then either \(G\) is \(n\)-permutable or \(G\) has finite order bounded by a function of \(m\) and \(n\).

MSC:

20E34 General structure theorems for groups
20F24 FC-groups and their generalizations
20F05 Generators, relations, and presentations of groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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