Show the centralisers $C_{S_6}(s)$ and $C_{S_6}(t)$ are isomorphic to $S_{3} \times C_{3}$

Let $s=(123)$ and $t=(123)(456)$ be elements of $S_6$. Show the centralisers $C_{S_6}(s)$ and $C_{S_6}(t)$ are isomorphic to $S_{3} \times C_{3}$.

My thoughts: Using the definition, $C_{S_6}(s)=\{g \in S_{6} : g^{-1}sg=s\}=\{g \in S_6 : (g(1) g(2) g(3))=(123)\}$. So we need to identify the elements $g \in S_{6}$ which leave $(123)$ intact (but may have any effect on 4,5,6). But I don't see how to identify this with $S_{3} \times C_{3}$.

With $t=(123)(456)$ this is even more confusing to me. How do I show this isomorphism?

Thanks a lot

Solution 1:

Hints. Let $A=\langle(123)(456)\rangle$, $B=\langle(14)(25)(35),(123)(465)\rangle$. Then

1. $C(t)=AB$;
2. $A\cong C_3$, $B\cong S_3$;
3. $AB\cong C_3\times S_3$.