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This problem is taken from I.N. Herstein

If $G$ is a group and $a \in G$ if of finite order and has only a finite number of conjugates in $G$, prove that these conjugates of $a$ generate a finite normal subgroup of $G$.

I would like to see a solution for this and also I would like to know whether the conjugates always generate a normal subgroup.

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Is this from a book by Herstein? If so which one? – Robin Chapman Aug 15 '10 at 19:17
@The Community: Please see this post on meta regarding the phrasing of questions and discuss (over there). – Tom Stephens Aug 15 '10 at 19:24
@Robin Chapman: Yes sir, its from "Topics in Algebra" second edition, supplementary problem section at the end of chapter two. – anonymous Aug 15 '10 at 19:24
@Chandru1: Thanks for adding a reference. Please continue to do so. – ShreevatsaR Aug 15 '10 at 19:25
The subgroup generated by an element of a group and its conjugates is always normal. – Robin Chapman Aug 15 '10 at 19:38
up vote 2 down vote accepted

This called Dicman's lemma and is 14.5.7 on page 425 (1st ed) or page 442 (2nd ed) of Robinson's Course in the Theory of Groups.

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