Characteristically complemented subgroup

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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

Symbol-free definition

A subgroup of a group is termed a characteristically complemented subgroup or characteristic retract if it satisfies the following equivalent conditions:

1. There is a retraction (viz an idempotent endomorphism) on the group, whose image is that subgroup, and whose kernel is a characteristic subgroup.
2. There is a characteristic subgroup that is a permutable complement to it.
3. There is a characteristic subgroup that is a lattice complement to it.

The corresponding characteristic subgroup (there may be more than one such) is termed a complemented characteristic subgroup.

Relation with other properties

Stronger properties

• Characteristically complemented characteristic subgroup
• Characteristically complemented normal subgroup

Weaker properties

• Quasicharacteristic retract
• Retract
• Permutably complemented subgroup
• Lattice-complemented subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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If $H \le K \le G$ are groups such that $H$ is characteristically complemented in $K$ and $K$ is characteristically complemented in $G$, then $H$ is characteristically complemented in $G$. This follows essentially from the fact that characteristicity is quotient-transitive. For full proof, refer: Characteristically complemented is transitive

Further information: Characteristicity is quotient-transitive, Quotient-transitive and stronger than normality implies complementary property is transitive

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties