Pronormal subgroup
From Encyclopedia of Mathematics
A subgroup of a group
satisfying the following condition: If
is a subgroup in
conjugate with
, then
is conjugate with
in the subgroup generated by
and
(cf. Conjugate elements). Sylow subgroups in finite groups, as well as Hall and Carter subgroups in finite solvable groups, are pronormal (cf. Sylow subgroup; Hall subgroup; Carter subgroup). The concept of a pronormal subgroup is closely connected with that of an abnormal subgroup. Every abnormal subgroup is pronormal, and the normalizer of a pronormal subgroup (cf. Normalizer of a subset) is abnormal.
References
[1] | L.A. Shemetkov, "Formations of finite groups" , Moscow (1978) (In Russian) |
Comments
References
[a1] | D.J.S. Robinson, "A course in the theory of groups" , Springer (1982) |
How to Cite This Entry:
Pronormal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pronormal_subgroup&oldid=33148
Pronormal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pronormal_subgroup&oldid=33148
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article