I am trying to find the normalizer of $\langle(123), (456)\rangle\ \subset S_6$. Then only way I can think of is to check element by element whether $g \in S_6$ takes the $(123), (456)$ back to the subgroup generated by them . But that seems to be a mission impossible since $S_6$ is to "large".
Normalizer of $\langle(123), (456)\rangle\ \subset S_6$
abstract-algebragroup-theory
Best Answer
Note that the normalizer of $\langle (123),(456)\rangle$ in $S_6$ has to preserve the partition $123\mid456$, so it is a subgroup of $S_3\operatorname{wr}C_2$. Clearly it is normalized by $S_3\times S_3$, so we need to check whether $(14)(25)(36)$ normalizes this subgroup, and it clearly does. So the normalizer is $S_3\operatorname{wr}C_2$.