1. ## Cycle permutation problem

Hello there. I have this two cycle permutation problems to prove,

1. $\displaystyle \left ( a b \right )^{-1} = \left ( a b \right )$

2. $\displaystyle (1,2,3, .... ,n)^{-1} = (n,n-1, ... ,2,1)$

Any idea?

2. ## Re: Cycle permutation problem

#1. What is $\displaystyle (a\ b)(a\ b)$?

#2. Use #1 and induction. (Note: $\displaystyle (1\ 2\ \cdots\ n\ n+1)=(1\ n+1)(1\ 2\ \cdots\ n)$.)

3. ## Re: Cycle permutation problem

Originally Posted by Nehushtan
#1. What is $\displaystyle (a\ b)(a\ b)$?

#2. Use #1 and induction. (Note: $\displaystyle (1\ 2\ \cdots\ n\ n+1)=(1\ n+1)(1\ 2\ \cdots\ n)$.)
1. It is decomposition into disjoint cycle?
2. I tried to solve it like this.
$\displaystyle \tau = (a_{1} a_{2} ... a_{n} )$
$\displaystyle \tau(a_{i})=a_{i+1}$
$\displaystyle \tau^{-1}(a_{i+1})=a_{i}$
$\displaystyle \tau: a_{1} \mapsto a_{2} \mapsto ...\mapsto a_{n}$
$\displaystyle \tau^{-1}: a_{n} \mapsto a_{n-1} \mapsto ...\mapsto a_{1}$
- Its obvious that tau^(-1), is just tau written in reverse.