Weakly procharacteristic 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]

Definition

Definition with symbols

A subgroup H of a group G is termed weakly procharacteristic if for any automorphism \sigma of G, the following holds: if K denotes the closure of H under the action of the cyclic group generated by \sigma, there exists g \in K such that \sigma(H) = gHg^{-1}.

Relation with other properties

Stronger properties

Weaker properties

Effect of property operators

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

The intermediately operator

Applying the intermediately operator to this property gives: intermediately automorph-conjugate subgroup

If H is a subgroup of G that is weakly procharacteristic in every intermediate subgroup of G containing it, then H is an intermediately automorph-conjugate subgroup of G. Conversely, if H is intermediately automorph-conjugate in G, then H is weakly procharacteristic in every intermediate subgroup.

Facts