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Difference between revisions of "CF-group"

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A [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120110/c1201101.png" /> in which every subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120110/c1201102.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120110/c1201103.png" /> is boundedly core-finite, abbreviated BCF, if there is an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120110/c1201104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120110/c1201105.png" /> has order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120110/c1201106.png" /> for all subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120110/c1201107.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120110/c1201108.png" />.)
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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.)
 
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.)
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====References====
 
====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>
 
<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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Latest revision as of 22:07, 14 January 2017

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
This article was adapted from an original article by M.R. Dixon (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article