# Cycle permutation problem

• March 13th 2013, 12:31 PM
DonnieDarko
Cycle permutation problem
Hello there. I have this two cycle permutation problems to prove,

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

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

Any idea?
• March 13th 2013, 06:22 PM
Nehushtan
Re: Cycle permutation problem
#1. What is $(a\ b)(a\ b)$?

#2. Use #1 and induction. (Note: $(1\ 2\ \cdots\ n\ n+1)=(1\ n+1)(1\ 2\ \cdots\ n)$.)
• March 14th 2013, 12:31 PM
DonnieDarko
Re: Cycle permutation problem
Quote:

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

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

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