Finding a permutation $a$ such that $a^{-1}(12)(34)a=(56)(13)$

Question

The question is to find the permutation $a $ such that $a^{-1}xa=y$ where $x=(12)(34) $ and $y=(56)(13)$

I found some answers to this question in this site but those don't clarify my doubt.

This is my doubt: $x=aya^{-1}$ $$x=(a(y))$$ $$(12)(34)=(a(5) a(6))(a(1) a(3))$$ So basically $a$ maps 5 to 1, 6 to 2, 1 to 3, 3 to 4.

Thus $$a: \begin{pmatrix} 1&2&3&4&5&6\\ 3&C&4&D&1&2 \end{pmatrix}$$

The possibilities for $C$ and $D$ are 5 and 6.

For any of those two cases for $C$ and $D$, I don't get the result which satisfies the given condition that $a^{-1}xa=y$

But if we take the inverse of this $a$ in the place of $a$ we get the result. And all the other answers in this site says a as stated below and use $a^{-1}$ in the place of $a$. I don't get the logic!! Can anyone check and tell me what I'm missing. (Please don't mark this as duplicate. I've seen similar questions and the answers of none of them answer my question.)

Answer 1

• "For any of those two cases for $C$ and $D$, I don't get the result which satisfies the given condition that $a^{-1}xa=y$"

  Well, you should:

  • For $(C,D)=(5,6),$ $$a^{-1}xa=(152643)(12)(34)(134625)=(13)(56)=y.$$
  • For $(C,D)=(6,5),$ $$a^{-1}xa=(1543)(26)(12)(34)(1345)(26)=(13)(56)=y.$$

• "But if we take the inverse of this $a$ in the place of $a$ we get the result."

  No, we don't:

  • For $(C,D)=(5,6),$ $$axa^{-1}=(134625)(12)(34)(152643)=(35)(46)\ne y.$$
  • For $(C,D)=(6,5),$ $$axa^{-1}=(1345)(26)(12)(34)(1543)(26)=(36)(45)\ne y.$$

• Note that these are not the only possible choices for $a:$ $$(12)(34)=(a(5) a(6))(a(1) a(3))$$ will also be achieved if, for instance, $a$ maps $(5,6,1,3)$ to $(2,1,4,3),$ or to $(3,4,2,1).$