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Let $G$ be a group. We can define

$F(G)=\{g\in G\,|\,o(\operatorname{Cl}(g))<\infty\},$

where $o(\operatorname{Cl}(g))$ is the order (cardinality) of the conjugacy class of $g$ in $G$. This set is a subgroup of $G$ because

1) $o(\operatorname{Cl}(1))=1<\infty;$

2) we have $g(xy)g^{-1}=(gxg^{-1})(gyg^{-1}),$ so $\operatorname{Cl}(xy)\subseteq\operatorname{Cl}(x)\operatorname{Cl}(y).$ Therefore, if $o(\operatorname{Cl}(x))<\infty\text{ and }o(\operatorname{Cl}(y))<\infty,$ then $o(\operatorname{Cl}(xy))<\infty.$

$F(G)$ is a characteristic subgroup of $G.$ Indeed, let $\alpha$ be an automorphism of $G$. Then

$\alpha(g)\alpha(x)(\alpha(g))^{-1}=\alpha(gxg^{-1})$

for all $g,x\in G.$ We have $o(\alpha(\operatorname{Cl}(x)))=o(\operatorname{Cl}(x))$ because $\alpha$ is a bijection and so, since $\{\alpha(g)\,|\,g\in G\}=G,$ we obtain

$o(\operatorname{Cl}(\alpha(g)))<\infty,$

and therefore

$\alpha(g)\in F(G).$

I would like to ask if there is a name for this characteristic subgroup. Also, since $F(G)$ is a piece of ad hoc notation, I'd be grateful if you could tell me what the common notation is.

Is the group $G/F(G)$ important? What is it called?

And finally, where can I read about it?

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    @SteveD Thank you. I know about this book but it's difficult to get where I live. The only available copy is in a library from which I don't have a right to borrow books. I've already used it a bit though and it's great.2012-02-12

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Yes, this is called the FC-centre of the group. I've seen $FC(G)$ used for this. I don't recall whether the quotient $G/FC(G)$ (or $G/F(G)$ in your notation) has a special name. There is a (nice) book on the subject by M. J. Tomkinson.

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    @ymar You're welcome. If you can't put your hands on a copy of Tomkinson, there is a little bit on this in D.J.S. Robinson's text.2012-02-14