I have found this topic here: How to find all the permutations which commute with a certain one, but I still could not understand it. Could anyone please explain how to think about doing it?
If I have (12)(34), working in the symmetric group 4 for instance, how would I find every possible permutation that permutes with it?
Thanks.
You want all the permutations $k \in S_4$ such that $$k(12)(34)k^{-1}=(12)(34).$$ Since $$ k(12)(34)k^{-1} =k(12)\color{red}k^{-1} \, \color{\red}k(34)k^{-1} = (k(1) \,\, k(2)) \,\,\,\, (k(3) \,\, k(4)) $$ Where $k(i)$ means the image of $i$ under permutation $k$.
You want this to be $$(k(1) \,\, k(2)) \,\,\,\, (k(3) \,\, k(4)) = (1 \,\, 2) \,\,\, (3\,\, 4)$$
So our permutation $k$ can be such that $$(k(1) \,\, k(2))=(1 \,\, 2) \quad \text{ and } \quad (k(3) \,\, k(4))=(3\,\, 4)$$ OR $$(k(1) \,\, k(2))=(3 \,\, 4) \quad \text{ and } \quad (k(3) \,\, k(4))=(1\,\, 2)$$
From the first scenario, we get either $k(1)=1,k(2)=2,k(3)=3,k(4)=4$ (identity permutation) or $k(1)=1,k(2)=2,k(3)=4,k(4)=3$ (this refers to the permutation $(34)$) or.....
Hope you can take it from here.
Added response:
I am adding this to answer the question you have raised in your comment.
Observe from the second scenario I have mentioned in my answer above, we can have $k(1)=3, k(2)=4$ and $k(3)=2, k(4)=1$. So ask yourself, what will be the cycle notation for this permutation?
we have ($1 \xrightarrow{k} 3 \xrightarrow{k} 2 \xrightarrow{k} 4 \xrightarrow{k} 1$) and this corresponds to $(1324)$.