Local subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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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

Relation with other properties

Stronger properties

• Local subgroup for a prime

Weaker properties

• Normalizer subgroup

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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A local subgroup of a group is also a local subgroup in any intermediate subgroup -- the solvable subgroup remains the same.