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Difference between revisions of "Pronormal subgroup"

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A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075410/p0754101.png" /> of a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075410/p0754102.png" /> satisfying the following condition: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075410/p0754103.png" /> is a subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075410/p0754104.png" /> conjugate with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075410/p0754105.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075410/p0754106.png" /> is conjugate with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075410/p0754107.png" /> in the subgroup generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075410/p0754108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075410/p0754109.png" /> (cf. [[Conjugate elements|Conjugate elements]]). Sylow subgroups in finite groups, as well as Hall and Carter subgroups in finite solvable groups, are pronormal (cf. [[Sylow subgroup|Sylow subgroup]]; [[Hall subgroup|Hall subgroup]]; [[Carter subgroup|Carter subgroup]]). The concept of a pronormal subgroup is closely connected with that of an [[Abnormal subgroup|abnormal subgroup]]. Every abnormal subgroup is pronormal, and the normalizer of a pronormal subgroup (cf. [[Normalizer of a subset|Normalizer of a subset]]) is abnormal.
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A subgroup $H$ of a [[Group|group]] $G$ satisfying the following condition: If $K$ is a subgroup in $G$ conjugate with $H$, then $K$ is conjugate with $H$ in the subgroup generated by $H$ and $K$ (cf. [[Conjugate elements|Conjugate elements]]). Sylow subgroups in finite groups, as well as Hall and Carter subgroups in finite solvable groups, are pronormal (cf. [[Sylow subgroup|Sylow subgroup]]; [[Hall subgroup|Hall subgroup]]; [[Carter subgroup|Carter subgroup]]). The concept of a pronormal subgroup is closely connected with that of an [[Abnormal subgroup|abnormal subgroup]]. Every abnormal subgroup is pronormal, and the normalizer of a pronormal subgroup (cf. [[Normalizer of a subset|Normalizer of a subset]]) is abnormal.
  
 
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Latest revision as of 10:01, 27 August 2014

A subgroup $H$ of a group $G$ satisfying the following condition: If $K$ is a subgroup in $G$ conjugate with $H$, then $K$ is conjugate with $H$ in the subgroup generated by $H$ and $K$ (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=12405
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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