permutations and disjoint cycles, the inverse.

Question

What is the inverse to the permutation $A = (2 4 6)(3 5)$ in $S_{6}$?

I got the inverse as $(2 5 6 4)$, but it's wrong?

Answer 1

To find the inverse of a permutation you simply reverse the cycles.

So, given $A=(2\ 4\ 6)(3\ 5)$; we have that $$A^{-1}=(6\ 4\ 2)(5\ 3).$$ Note that the inverse of a $2$-cycle is just the $2$-cycle itself.

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Observe that $A=(2\ 4\ 6)(3\ 5)$ in its two line notation form is \begin{equation} A= \left(\begin{array}{cccc} 1\ 2\ 3\ 4\ 5\ 6 \\ 1\ 4\ 5\ 6\ 3\ 2\\ \end{array}\right). \end{equation}

Now, to find the inverse; we take the bottom row and put it in ascending order and see what we get. So \begin{equation} A^{-1}= \left(\begin{array}{cccc} 1\ 2\ 3\ 4\ 5\ 6 \\ 1\ 6\ 5\ 2\ 3\ 4\\ \end{array}\right), \end{equation}

which, in disjoint cycle notation, corresponds to $$(1)(2\ 6\ 4)(3\ 5).$$