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Groups containing an element commuting with a finite number of its conjugates. (English. Russian original) Zbl 0941.20039
Algebra Logika 37, No. 6, 637-650 (1998); translation in Algebra Logic 37, No. 6, 363-370 (1998).
Let \(G\) be a group and let \(a\) be an element of \(G\) such that the set \(C_G(a)\cap a^G\) is finite. Let \(\Gamma\) be the graph such that the vertex set \(V(\Gamma)\) of \(\Gamma\) is the conjugacy class \(a^G\) and the edge set \(E(\Gamma)\) of \(\Gamma\) is the set of all pairs \(\{x,y\}\) such that \(x\neq y\) and \(xy=yx\). Then \(\Gamma\) is a locally finite graph. Let \(\Gamma^0\) be a connected component of \(\Gamma\) and let \(H=\langle V(\Gamma^0)\rangle\) be the group generated by \(V(\Gamma)^0\). The main result of the article under review is as follows. Theorem. Let \(G\) be a group, let \(\Gamma\) be a locally finite graph, let \(\Gamma^0\) be a connected component of \(\Gamma\), and let \(H=\langle V(\Gamma^0)\rangle\). If every pair of vertices of \(\Gamma^0\) generates a nilpotent subgroup (finite nilpotent subgroup, finite \(p\)-group) then \(H\) is a locally nilpotent group (torsion nilpotent group, locally finite \(p\)-group).

20F24 FC-groups and their generalizations
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20E26 Residual properties and generalizations; residually finite groups
20E07 Subgroup theorems; subgroup growth
20F18 Nilpotent groups
20F50 Periodic groups; locally finite groups
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