# Definition:Weakly Pronormal Subgroup/Definition 1

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## Definition

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

$H$ is **weakly pronormal in $G$** if and only if:

- $\forall g \in G: \exists x \in H^{\gen g}: H^x = H^g$

where:

- $H^{\gen g}$ denotes the smallest subgroup of $G$ containing $H$, generated by the conjugacy action by the cyclic subgroup of $G$ generated by $g$
- $H^x$ denotes the conjugate of $H$ by $x$.