1. ## Permutations & Cycles

In $S_6$, let $\beta = (61)(65)(153)(653)$

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

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

2. In Cycle notation, 1-cycles are often omitted, so you can right
$\beta = (1)(2)(356)(4) = (356)$
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 $\beta$ into transpositions (2-cycles)

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

3. Originally Posted by Haven
In Cycle notation, 1-cycles are often omitted, so you can right
$\beta = (1)(2)(356)(4) = (356)$
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 $\beta$ into transpositions (2-cycles)

so, $\beta = (36)(35)$
Since there are 2 transpositions, $\beta$ is an even permutation
Of course, 1 cycles just send the number to itself. Thanks for the reply, appreciate it.