Subnormalizer subset

Definition

Suppose $H$ is a subgroup of a group $G$. The subnormalizer subset (sometimes termed the subnormalizer) of $H$ in $G$ is defined as:

$\{g\in G\mid H\triangleleft \triangleleft \langle H,g\rangle \}$.

The term subnormalizer is sometimes reserved for the situation where this subset is a subgroup and $H$ is subnormal in it, in which case the original subgroup $H$ is termed a subgroup having a subnormalizer. Note that the subnormalizer subset need not be a subgroup because subnormality is not upper join-closed.

References

• Subnormal subgroups of groups by John C. Lennox and Stewart E. Stonehewer, Oxford Mathematical Monographs, ISBN 019853552X, Page 238, Section 7.7 (The subnormalizer of a subgroup), ^{More info}