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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).

##### MSC:
 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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