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Groups preserving the cardinality of subsets product under permutations. (English) Zbl 0864.20016

A group \(G\) has the property \(\text{PC}(2,n)\) (where \(n>1\)) if either \(|G|=1\) or else, for each \(n\)-tuple of 2-element subsets \((X_1,X_2,\dots,X_n)\) of \(G\), there exists \(\sigma\neq 1\) in \(S_n\) such that \(|X_1\cdots X_n|=|X_{\sigma(1)}\cdots X_{\sigma(n)}|\). Also \(G\) has \(\text{PC}(2)\) if it has some \(\text{PC}(2,n)\).
It is shown that a group \(G\) has \(\text{PC}(2,3)\) if and only if it is either abelian or a Hamiltonian 2-group. The author also proves that if \(G\) has \(\text{PC}(2)\), then \(G\) is centre-by-finite exponent \(f(n)\) for some function \(f\). Some special types of groups with \(\text{PC}(2)\) are characterized.

MSC:

20E34 General structure theorems for groups
20F05 Generators, relations, and presentations of groups
20F24 FC-groups and their generalizations
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References:

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