Let (Formula presented.) be a group-word. For a group (Formula presented.), let (Formula presented.) denote the set of all (Formula presented.)-values in (Formula presented.) and let (Formula presented.) denote the verbal subgroup of (Formula presented.) corresponding to (Formula presented.). The group (Formula presented.) is an (Formula presented.)-group if the set of conjugates (Formula presented.) is finite for all (Formula presented.). It is known that if (Formula presented.) is a concise word, then (Formula presented.) is an (Formula presented.)-group if and only if (Formula presented.) is (Formula presented.)-embedded in (Formula presented.), that is, the conjugacy class (Formula presented.) is finite for all (Formula presented.). There are examples showing that this is no longer true if (Formula presented.) is not concise. In the present paper, for an arbitrary word (Formula presented.), we show that if (Formula presented.) is an (Formula presented.)-group, then the commutator subgroup (Formula presented.) is (Formula presented.)-embedded in (Formula presented.). We also establish the analogous result for (Formula presented.)-groups, that is, groups in which the sets (Formula presented.) are boundedly finite.

### ON GROUPS WITH FINITE CONJUGACY CLASSES IN A VERBAL SUBGROUP

#### Abstract

Let (Formula presented.) be a group-word. For a group (Formula presented.), let (Formula presented.) denote the set of all (Formula presented.)-values in (Formula presented.) and let (Formula presented.) denote the verbal subgroup of (Formula presented.) corresponding to (Formula presented.). The group (Formula presented.) is an (Formula presented.)-group if the set of conjugates (Formula presented.) is finite for all (Formula presented.). It is known that if (Formula presented.) is a concise word, then (Formula presented.) is an (Formula presented.)-group if and only if (Formula presented.) is (Formula presented.)-embedded in (Formula presented.), that is, the conjugacy class (Formula presented.) is finite for all (Formula presented.). There are examples showing that this is no longer true if (Formula presented.) is not concise. In the present paper, for an arbitrary word (Formula presented.), we show that if (Formula presented.) is an (Formula presented.)-group, then the commutator subgroup (Formula presented.) is (Formula presented.)-embedded in (Formula presented.). We also establish the analogous result for (Formula presented.)-groups, that is, groups in which the sets (Formula presented.) are boundedly finite.
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2017
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11386/4702697`
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