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If a group $G$ is generated by finitely many subgroups $G_i$ and $H$ a subgroup of $G$, under which conditions can $N_G(K)$, the normalizer of $K$ in $G$, be generated by all the normailizers $N_{G_i}(K)$?

More concretely, let $G$ be a compact connected Lie group and $K$ a finite abelian subgroup of $G$ whose centralizer in $G$ is $K$, i.e., $K$ is a maximal finite abelian subgroup of $G$. Assume that $G$ is generated by closed connected abelian subgroups $G_i$, $i=1\cdots n$. (Each $G_i$ is a torus.) It is known that $K$ normailizes each $G_i$. Is $N_G(K)$, the normalizer of $K$ in $G$, generated by all the normailizers $N_{G_i}(K)$,$i=1\cdots n$ ?

This is true for all the examples I know. But I cannot find a proof. Your help is very appreciated!

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