# Procharacteristic subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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

• (Left-action convention): A subgroup $H$ of a group $G$ is termed procharacteristic in $G$ if, for any automorphism $\sigma$ of $G$, there exists $g \in \langle H, \sigma(H) \rangle$ such that $gHg^{-1} = \sigma(H)$.
• (Right-action convention): A subgroup $H$ of a group $G$ is termed procharacteristic in $G$ if, for any automorphism $\sigma$ of $G$, there exists $g \in \langle H, H^\sigma\rangle$ such that $H^g = H^\sigma$.

## Facts

• Any procharacteristic subgroup of a normal subgroup is pronormal. Further information: Procharacteristic of normal implies pronormal
• A subgroup $H$ of a group $G$ is procharacteristic in $G$ if and only if whenever $G$ is normal in some group $K$, $H$ is pronormal in $K$. Further information: Left residual of pronormal by normal is procharacteristic