Visit

Groupprops, The Group Properties Wiki (pre-alpha)
Take a short survey about Math Resources on the Internet.

From Groupprops

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW:
VIEW RELATED:
Learn more about terminology local to the wiki | view a complete list of such terminology

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions[SHOW MORE]

VIEW RELATED:
VIEW FACTS THAT:|
RANDOM TIP:The metaproperties section lists important facts about the subgroup property, and addresses many of the natural questions that arise about it. It has links to proofs.

Definition

Definition with symbols

A subgroup H of a group G is termed a subnormal stability automorphism-invariant subgroup if, for any subnormal series of G, and any stability automorphism σ of that subnormal series, σ(H) = H.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Cofactorial automorphism-invariant subgroup for a finite group.

Weaker properties

• Normal stability automorphism-invariant subgroup

Facts

• For a finite group, the stability automorphisms of any subnormal series have no other prime factors to their order than the prime factors of the order of the group. For full proof, refer: Stability group of subnormal series of finite group has no other prime factors
• For a group of prime power order, the stability automorphisms of subnormal series are precisely the same as the p-automorphisms. For full proof, refer: Stability group of subnormal series of p-group is p-group, p-group of automorphisms of p-group is contained in stability group of some normal series