The Group Properties Wiki (pre-alpha)

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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.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: | Subgroup property non-implications | | | |
RANDOM TIP:The relation with other properties section lists stronger and weaker properties, along with links to proofs of the implications and non-implications. This helps give a feel of how the subgroup property relates to other properties.

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 ascendant closure operator 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 2.3 Opposites 3 Metaproperties 3.1 Transitivity 3.2 Trimness 3.3 Intermediate subgroup condition 3.4 Intersection-closedness

Definition

Symbol-free definition

A subgroup of a group is said to be ascendant if there is an ascending series of subgroups indexed by ordinals, each normal in the next, that starts at the given subgroup, and terminates at the whole group.

Definition with symbols

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

• H₀ = H
• (viz H_α is a normal subgroup of H_{α + 1})
• 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 ascendant closure operator

The subgroup property of being an ascendant subgroup is obtained by applying the ascendant 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

Opposites

• Self-normalizing subgroup

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

An ascendant subgroup of an ascendant subgroup is ascendant. The proof relies on simply concatenating the two ascending 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

Thetrivial subgroup and the whole group are both normal subgroups, hence they are also both ascendant 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 ascendant subgroup of a group is also ascendant in every intermediate subgroup. The proof of this follows by intersecting every member of the ascending 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 ascendant subgroups is ascendant. This again follows by intersecting the ascending series for each subgroup.