Intermediately normality-large subgroup

From Groupprops
Jump to: navigation, search
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]


Symbol-free definition

A subgroup of a group is termed intermediately normality-large if it satisfies the following equivalent conditions:

  1. It is normality-large in every intermediate subgroup
  2. It does not occur as a retract of any subgroup properly containing it.

Definition with symbols

A subgroup H of a group G is termed intermediately normality-large in G if it satisfies the following equivalent conditions:

  1. For any subgroup K of G containing H, and any normal subgroup N of K such that N \cap H is trivial, N must be trivial
  2. If K is a subgroup of G containing H, and \alpha is a retraction from K to H (i.e. a surjective homomorphism that restricts to the identity map on H), then K = H.