Strongly paranormal subgroup

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed strongly paranormal in $G$ if for any $g\in G$, the following identity holds:

$[[g,H],H]=[g,H]$

Here by $[g,H]$ we mean the subgroup generated by all commutators between $g$ and elements of $H$.

Relation with other properties

Stronger properties

• Central subgroup

Weaker properties

• Strongly polynormal subgroup
• Paranormal subgroup
• Polynormal subgroup
• Weakly normal subgroup
• Intermediately subnormal-to-normal subgroup
• Subnormal-to-normal subgroup

References

• On the lattice of subgroups by Z. I. Borevich and O. N. Macedonska, Zap. Nauchn. Semin., LOMI 101, 13-19, 1980