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 defining ingredient::subnormal subgroup of $$K$$, and further, if $$H \le L \le G$$ is such that $$H$$ is subnormal in $$L$$, then $$L \le K$$.

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.