Characteristically DP-decomposable subgroup

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

Definition with symbols

A subgroup $H$ inside a group $G$ is said to be characteristically DP-decomposable if we can express $G$ as an internal direct product of subgroups $G_w$ such that $H$ is the internal direct product of all the $H_w=H\cap G_w$, and further, each $H_w$ is characteristic in $G_w$.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Direct factor
• CDF-subgroup

Weaker properties

• Normal subgroup

Metaproperties

Direct product-closedness

This subgroup property is direct product-closed: it is closed under taking arbitrary direct products of groups

This actually follows from the way it's defined. The given property is the strongest direct product-closed subgroup property that is weaker than the property of being characteristic.

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 other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Clearly, the trivial subgroup, as well as the whole group, are characteristically DP-decomposable.