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Definition: A subgroup $H$ of $G$ is said to be pronormal in $G$ if for all $g\in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x =H^g$.

Definition: A subgroup $H$ of $G$ is said to be weakly pronormal in $G$ if for $g \in G$, there exists $x \in H^{\langle g \rangle} := \langle g^nHg^{-n} \, |\, n\in \mathbb{Z} \rangle$ such that $H^x =H^g$.

By the inclusion $\langle H, H^g \rangle \leq H^{\langle g \rangle}$, we have that pronormality implies weak pronormality. It is also known that they coincide for finite solvable groups.

I need to construct an example to show that weak pronormality does not imply pronormality in general. This would have to be a non-solvable group to start with.

Definition: A subgroup $H$ of $G$ is said to be pronormal in $G$ if for all $g\in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x =H^g$.

Definition: A subgroup $H$ of $G$ is said to be weakly pronormal in $G$ if for $g \in G$, there exists $x \in H^{\langle g \rangle} := \langle g^nHg^{-n} \, |\, n\in \mathbb{Z} \rangle$ such that $H^x =H^g$.

By the inclusion $\langle H, H^g \rangle \leq H^{\langle g \rangle}$, we have that pronormality implies weak pronormality. It is also known that they coincide for finite solvable groups. However it does not hold in general according to the paper

I need to construct an example to show that weak pronormality does not imply pronormality for finite groups in general. This would have to be a non-solvable finite group to start with.

Definition: A subgroup $H$ of $G$ is said to be pronormal in $G$ if for all $g\in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x =H^g$.

Definition: A subgroup $H$ of $G$ is said to be weakly pronormal in $G$ if for $g \in G$, there exists $x \in H^{\langle g \rangle} := \langle g^nHg^{-n} \, |\, n\in \mathbb{N} \rangle$ such that $H^x =H^g$.

By the inclusion $\langle H, H^g \rangle \leq H^{\langle g \rangle}$, we have that pronormality implies weak pronormality. It is also known that they coincide for finite solvable groups.

I need to construct an example to show that weak pronormality does not imply pronormality in general. This would have to be a non-solvable group to start with.

Definition: A subgroup $H$ of $G$ is said to be pronormal in $G$ if for all $g\in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x =H^g$.

Definition: A subgroup $H$ of $G$ is said to be weakly pronormal in $G$ if for $g \in G$, there exists $x \in H^{\langle g \rangle} := \langle g^nHg^{-n} \, |\, n\in \mathbb{Z} \rangle$ such that $H^x =H^g$.

By the inclusion $\langle H, H^g \rangle \leq H^{\langle g \rangle}$, we have that pronormality implies weak pronormality. It is also known that they coincide for finite solvable groups.

I need to construct an example to show that weak pronormality does not imply pronormality in general. This would have to be a non-solvable group to start with.

Definition: A subgroup $H$ of $G$ is said to be pronormal in $G$ if for all $g\in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x =H^g$.

Definition: A subgroup $H$ of $G$ is said to be weakly pronormal in $G$ if for $g \in G$, there exists $x \in H^{\langle g \rangle} := \langle gHg^{-1} | g\in G \rangle$ such that $H^x =H^g$.

By the inclusion $\langle H, H^g \rangle \leq H^{\langle g \rangle}$, we have that pronormality implies weak pronormality. It is also known that they coincide for finite solvable groups.

I need to construct an example to show that weak pronormality does not imply pronormality in general. This would have to be a non-solvable group to start with.

Definition: A subgroup $H$ of $G$ is said to be pronormal in $G$ if for all $g\in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x =H^g$.

Definition: A subgroup $H$ of $G$ is said to be weakly pronormal in $G$ if for $g \in G$, there exists $x \in H^{\langle g \rangle} := \langle g^nHg^{-n} \, |\, n\in \mathbb{N} \rangle$ such that $H^x =H^g$.

By the inclusion $\langle H, H^g \rangle \leq H^{\langle g \rangle}$, we have that pronormality implies weak pronormality. It is also known that they coincide for finite solvable groups.

I need to construct an example to show that weak pronormality does not imply pronormality in general. This would have to be a non-solvable group to start with.