# Weakly pronormal 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]
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

## Definition

### Definition with symbols

A subgroup $H$ of a group $G$ is termed weakly pronormal or is said to satisfy the Frattini property if it satisfies the following equivalent conditions:

• Given any $g \in G$, there exists $x \in H^{\langle g \rangle}$ such that $H^x = H^g$. Here $H^{\langle g \rangle}$ denotes the smallest subgroup of $G$ containing $H$ which is closed under conjugation by $g$.
• If $H \le K \le L \le G$ are such that $K$ is a normal subgroup of $L$, we have $KN_L(H) = L$.

### Equivalence of definitions

Further information: Equivalence of definitions of weakly pronormal subgroup

## Relation with other properties

For a picture of related subnormal-to-normal subgroup properties, refer this implication diagram.