Local subgroup

Definition
A local subgroup of a group is defined as a subgroup that occurs as the normalizer of a nontrivial solvable subgroup. In symbols, a subgroup $$H$$ of $$G$$ is termed a local subgroup if there is a nontrivial solvable subgroup $$Q$$ of $$G$$ such that $$H = N_G(Q)$$.

Note that the nontriviality of $$Q$$ is crucial to the definition.

Facts

 * Local subgroup of finite group is contained in p-local subgroup for some prime p

Stronger properties

 * Local subgroup for a prime

Weaker properties

 * Normalizer subgroup

Metaproperties
A local subgroup of a group is also a local subgroup in any intermediate subgroup -- the solvable subgroup remains the same.