Subnormalizer

Definition

Suppose $H$ is a subgroup of a group $G$. A subnormalizer of $H$ in $G$ is a subgroup $K$ of $G$ containing $H$ such that $H$ is a subnormal subgroup of $K$, and further, if $H\leq L\leq G$ is such that $H$ is subnormal in $L$, then $L\leq K$.

Since subnormality is not upper join-closed, not every subgroup need have a subnormalizer.

The term subnormalizer is also sometimes used for the subnormalizer subset, which is the subset comprising all elements that subnormalize it. When a subnormalizer exists in the sense described here, it coincides with the subnormalizer subset; however, the subnormalizer subset always exists, while the subnormalizer (subgroup) need not.

Also, the subnormalizer subset may exist and be a subgroup, but it may not be a subnormalizer in this sense.