Direct power-closed characteristic subgroup

From Groupprops
Jump to: navigation, search
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]

Definition

A subgroup H of a group G is termed a direct power-closed characteristic subgroup if for every cardinal \alpha, the subgroup H^\alpha inside the direct power G^\alpha (using the external direct product) is a characteristic subgroup of G^\alpha.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
marginal subgroup follows from marginality is direct power-closed and marginal implies characteristic |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite direct power-closed characteristic subgroup |FULL LIST, MORE INFO
characteristic subgroup |FULL LIST, MORE INFO