Weakly pronormal subgroup
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Contents
Definition
Definition with symbols
A subgroup of a group is termed weakly pronormal or is said to satisfy the Frattini property if it satisfies the following equivalent conditions:
- Given any , there exists such that . Here denotes the smallest subgroup of containing which is closed under conjugation by .
- If are such that is a normal subgroup of , we have .
Equivalence of definitions
Further information: Equivalence of definitions of weakly pronormal subgroup
Relation with other properties
For a picture of related subnormal-to-normal subgroup properties, refer this implication diagram.
Stronger properties
- Normal subgroup
- Join-transitively pronormal subgroup
- Pronormal subgroup
- Abnormal subgroup
- Weakly abnormal subgroup
- Intermediately isomorph-conjugate subgroup
- Intermediately automorph-conjugate subgroup
- Sylow subgroup
- Sylow subgroup of normal subgroup: This follows from Sylow of normal implies pronormal and pronormal implies weakly pronormal.
- Intermediately isomorph-conjugate subgroup of normal subgroup
- Intermediately automorph-conjugate subgroup of normal subgroup
- Weakly procharacteristic subgroup
Weaker properties
- Polynormal subgroup
- Intermediately subnormal-to-normal subgroup
- Subnormal-to-normal subgroup
- Subgroup with weakly abnormal normalizer
- Subgroup with self-normalizing normalizer
References
Journal references
- Pronormality in finite groups by T. A. Peng, Journal of the London Mathematical Society, ISSN 14697750 (online), ISSN 00246107 (print), Volume 3, Page 301 - 306(Year 1971): ^{Weblink}^{More info}
- Transitivity of normality and pronormal subgroups by L. A. Kurdachenko and I. Ya. Subbotin, Combinatorial group theory, discrete groups, and number theory, Volume 421, Page 201 - 210(Year 2006): ^{}^{More info}
- Abnormal, pronormal, contranormal, and Carter subgroups in some generalized minimax groups by L. A. Kurdachenko, J. Otal and I. Ya. Subbotin, : ^{Weblink}^{More info}