View source for CF-group

A [[Group|group]] $G$ in which every subgroup of $G$ has finite index over its core (cf. also [[Core of a subgroup|Core of a subgroup]]). Locally finite CF-groups (cf. [[Locally finite group|Locally finite group]]) satisfying this condition are studied in detail in [[#References|[a1]]], where it is shown that every locally finite CF-group is Abelian-by-finite, and has the stronger property that it is boundedly core-finite. (A group $G$ is boundedly core-finite, abbreviated BCF, if there is an integer $n$ such that $H / \operatorname{core}_G (H)$ has order at most $n$ for all subgroups $H$ of $G$.)

H. Smith and J. Wiegold showed in [[#References|[a3]]] that a locally graded BCF group is Abelian-by-finite and that every nilpotent CF-group is BCF and Abelian-by-finite. (A group is called locally graded if every non-trivial finitely generated subgroup has a non-trivial finite image.)

There exist infinite simple two-generator groups with all proper non-trivial subgroups cyclic of prime order, so there exist CF-groups which are not Abelian-by-finite.

CF-groups are dual to the class of groups in which every subgroup has finite index in its normal closure. Such groups were studied in [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.T. Buckley,   J.C. Lennox,   B.H. Neumann,   H. Smith,   J. Wiegold,   "Groups with all subgroups normal-by-finite"  ''J. Austral. Math. Soc. (Ser. A)'' , '''59'''  (1995)  pp. 384–398</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.H. Neumann,   "Groups with finite classes of conjugate subgroups"  ''Math. Z.'' , '''63'''  (1955)  pp. 76–96</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Smith,   J. Wiegold,   "Locally graded groups with all subgroups normal-by-finite"  ''J. Austral. Math. Soc. (Ser. A)'' , '''60'''  (1996)  pp. 222–227</TD></TR></table>

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