Weakly pronormal subgroup

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed weakly pronormal or is said to satisfy the Frattini property if it satisfies the following equivalent conditions:

• Given any $g \in G$, there exists $x \in H^{\langle g \rangle}$ such that $H^x = H^g$. Here $H^{\langle g \rangle}$ denotes the smallest subgroup of $G$ containing $H$ which is closed under conjugation by $g$.
• If $H \le K \le L \le G$ are such that $K$ is a normal subgroup of $L$, we have $KN_L(H) = L$.

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): ^{WeblinkMore 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,  : ^{WeblinkMore info}