# 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 $\sigma(H) = gHg^{-1}$.

## Effect of property operators

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### 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.