Subgroup having a subnormalizer

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed a subgroup having a subnormalizer if $H$ has a subnormalizer in $G$: there exists a subgroup $K$ of $G$ containing $H$ such that $H$ is a subnormal subgroup of $K$, and if $H\leq L\leq G$ such that $H$ is subnormal in $L$, then $L\leq K$.

Not every subgroup has a subnormalizer, because subnormality is not upper join-closed.

Formalisms

In terms of the izer-existence operator

This property is obtained by applying the izer-existence operator to the property: subnormal subgroup
View other properties obtained by applying the izer-existence operator

Relation with other properties

Stronger properties

• Intermediately subnormal-to-normal subgroup
• Subnormal subgroup
• Normal subgroup
• Self-normalizing subgroup
• Pronormal subgroup
• Weakly pronormal subgroup
• Paranormal subgroup
• Polynormal subgroup
• Weakly normal subgroup