CC-group

A group $G$ which has Chernikov conjugacy classes. More precisely, let $x \in G$ and let $\langle x \rangle ^ { G }$ denote the normal closure of $\langle x \rangle$ in $G$. Then $G$ is a CC-group if $G / C _ { G } ( \langle x \rangle ^ { G } )$ is a Chernikov group for all $x \in G$. Such groups are generalizations of groups with finite conjugacy classes (cf. Locally normal group). CC-groups were first introduced by Ya.D. Polovitskii in [a3]. He showed that if $G$ is a CC-group, then the derived group of $G$ is locally finite and, in fact, locally normal and Chernikov.

Much of the theory established for FC-groups (see Group with a finiteness condition) has been generalized, to some extent or other, to the class of CC-groups. E.g., in [a1] J. Alcázar and J. Otal showed that the Sylow $p$-subgroups of a CC-group are locally conjugate. The theory of formations of groups and the theory of Fitting classes in finite solvable groups has also been extended to the class of locally solvable CC-groups (see [a2] for a survey).

References

[a1]	J. Alcázar, J. Otal, "Sylow subgroups of groups with Černikov conjugacy classes" J. Algebra , 110 (1987) pp. 507–513
[a2]	J. Otal, J.M. Peña, "Fitting classes and formations of locally soluble CC-groups" Boll. Un. Mat. Ital. B , 10 (1996) pp. 461–478
[a3]	Ya.D. Polovitskii, "Groups with extremal classes of conjugate elements" Sib. Mat. Zh. , 5 (1964) pp. 891–895 (In Russian)