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As this questionquestion was closed as a duplicate of Existence of simultaneously normal finite index subgroupsExistence of simultaneously normal finite index subgroups, I am opening a modified version here.

$G$ is a group, $A$ and $B$ are two subgroups of $G$. Suppose that $A∩B$ has finite index in both $A$ and $B$. It has been shown herehere that $A\cap B$ need not have a subgroup of finite index which is normal in both $A$ and $B$.

Question : If the set $A\cup B$ is normalised by $G$, does $A\cap B$ has a subgroup of finite index which is normal in both $A$ and $B$.

As this question was closed as a duplicate of Existence of simultaneously normal finite index subgroups, I am opening a modified version here.

$G$ is a group, $A$ and $B$ are two subgroups of $G$. Suppose that $A∩B$ has finite index in both $A$ and $B$. It has been shown here that $A\cap B$ need not have a subgroup of finite index which is normal in both $A$ and $B$.

Question : If the set $A\cup B$ is normalised by $G$, does $A\cap B$ has a subgroup of finite index which is normal in both $A$ and $B$.

As this question was closed as a duplicate of Existence of simultaneously normal finite index subgroups, I am opening a modified version here.

$G$ is a group, $A$ and $B$ are two subgroups of $G$. Suppose that $A∩B$ has finite index in both $A$ and $B$. It has been shown here that $A\cap B$ need not have a subgroup of finite index which is normal in both $A$ and $B$.

Question : If the set $A\cup B$ is normalised by $G$, does $A\cap B$ has a subgroup of finite index which is normal in both $A$ and $B$.

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Drike
Drike
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Finite indexed simultaneously normal subgroup in a normal union of two groups

As this question was closed as a duplicate of Existence of simultaneously normal finite index subgroups, I am opening a modified version here.

$G$ is a group, $A$ and $B$ are two subgroups of $G$. Suppose that $A∩B$ has finite index in both $A$ and $B$. It has been shown here that $A\cap B$ need not have a subgroup of finite index which is normal in both $A$ and $B$.

Question : If the set $A\cup B$ is normalised by $G$, does $A\cap B$ has a subgroup of finite index which is normal in both $A$ and $B$.