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From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
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VIEW RELATED: | Subgroup property non-implications | | | |
RANDOM SUBGROUP PROPERTY: Fully characteristic subgroup: A subgroup such that any endomorphism of the whole group takes the subgroup to within itself.

This is a variation of subnormality
View a complete list of variations of subnormality OR read a survey article on varying subnormality

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 1.3 In terms of the descendant closure operator 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 2.3 Related properties 2.4 Opposites 3 Facts 3.1 Descendant-contranormal factorization 4 Metaproperties 4.1 Transitivity 4.2 Trimness 4.3 Intermediate subgroup condition 4.4 Intersection-closedness

Definition

Symbol-free definition

A subgroup of a group is said to be descendant if there is a descending series of subgroups indexed by ordinals, each normal in its predecessor, that starts at the whole group, and terminates at the given subgroup.

Definition with symbols

A subgroup H of a group G is termed descendant if we have a series H_α for every ordinal α such that:

• H₀ = G
• (viz H_{α + 1} is a normal subgroup of H_α)
• There is some β such that H_β = G (note that beyond this point we must get all H_α = G).

In terms of the descendant closure operator

The subgroup property of being an descendant subgroup is obtained by applying the descendant closure operator to the subgroup property of being normal.

Relation with other properties

Stronger properties

• Normal subgroup
• Subnormal subgroup
• Hypernormalized subgroup

Weaker properties

• Serial subgroup

Related properties

• Ascendant subgroup

Opposites

• Contranormal subgroup

Facts

Descendant-contranormal factorization

This result states that given any subgroup H of G, there is a unique subgroup K containing H such that H is contranormal in K and K is descendant in G.

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property.
View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties

A descendant subgroup of a descendant subgroup is descendant. The proof relies on simply concatenating the two descending series.

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself)
View all trim subgroup properties OR view trivially true subgroup properties OR view identity-true subgroup properties

The trivial subgroup and the whole group are both normal subgroups, hence they are also both descendant subgroups.

Intermediate subgroup condition

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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the intermediate subgroup condition

Any descendant subgroup of a group is also descendant in every intermediate subgroup. The proof of this follows by intersecting every member of the descending series with the intermediate subgroup and observing that normality at each stage is preserved.

Intersection-closedness

This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property
View a complete list of intersection-closed subgroup properties

An arbitrary intersection of descendant subgroups is ascendant. This again follows by intersecting the ascending series for each subgroup.