# Permutations & Cycles

• Mar 12th 2010, 10:42 AM
craig
Permutations & Cycles
In $\displaystyle S_6$, let $\displaystyle \beta = (61)(65)(153)(653)$

I found the cycle decomposition of $\displaystyle \beta$ to be $\displaystyle (1)(2)(356)(4)$, the question then asks me what is the cycle length of $\displaystyle \beta$.

This is probably a stupid question but is it cycle length 4? I think I missed that small part of the lecture...

• Mar 12th 2010, 11:23 AM
Haven
In Cycle notation, 1-cycles are often omitted, so you can right
$\displaystyle \beta = (1)(2)(356)(4) = (356)$
So, we are left with a cycle of length 3 (since it permutes 3 numbers)

Now if we want to determine if this is an even or odd permutation, we must decompose $\displaystyle \beta$ into transpositions (2-cycles)

so,$\displaystyle \beta = (36)(35)$
Since there are 2 transpositions, $\displaystyle \beta$ is an even permutation
• Mar 12th 2010, 03:19 PM
craig
Quote:

Originally Posted by Haven
In Cycle notation, 1-cycles are often omitted, so you can right
$\displaystyle \beta = (1)(2)(356)(4) = (356)$
So, we are left with a cycle of length 3 (since it permutes 3 numbers)

Now if we want to determine if this is an even or odd permutation, we must decompose $\displaystyle \beta$ into transpositions (2-cycles)

so,$\displaystyle \beta = (36)(35)$
Since there are 2 transpositions, $\displaystyle \beta$ is an even permutation

Of course, 1 cycles just send the number to itself. Thanks for the reply, appreciate it.