# CF-group

A group $G$ in which every subgroup of $G$ has finite index over its core (cf. also Core of a subgroup). Locally finite CF-groups (cf. Locally finite group) satisfying this condition are studied in detail in [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 [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 [a2].

#### References

[a1] | 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 |

[a2] | B.H. Neumann, "Groups with finite classes of conjugate subgroups" Math. Z. , 63 (1955) pp. 76–96 |

[a3] | H. Smith, J. Wiegold, "Locally graded groups with all subgroups normal-by-finite" J. Austral. Math. Soc. (Ser. A) , 60 (1996) pp. 222–227 |

**How to Cite This Entry:**

CF-group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=CF-group&oldid=40185