Breaking a cycle into 2 non-disjoint cycles

Question

I have just started reading permutation groups and have gained a basic idea of the topic. I have a cycle of length $5$ given as $(32541)$ and I want to break it into two non-disjoint cycles. I know there can be many such cycles, but is there any way to find non-disjoint cycles of equal length(say $3$)?

Thanks!

Shaun · Accepted Answer · 2021-10-27T17:38:58.747

1

Sure:

$$\begin{align} (32541)&=\color{red}{(32)(25)}\color{blue}{(54)(41)}\\ &=\color{red}{(253)}\color{blue}{(154)}. \end{align}$$

edited Oct 27 '21 at 17:38

answered Oct 27 '21 at 17:33

Shaun

47,747

Yes, I can check that by performing group multiplication on RHS I will get $(32541)$ but how to get RHS at first place? for example why not $(541)(321)$ or any other combination? Thanks! – indmath Oct 27 '21 at 17:40
See here, @indmath. – Shaun Oct 27 '21 at 17:43
So basically we have to break it into transformations(as we do while checking even and odd permutaions) and then combine two of those combinations at a time? @Shaun – indmath Oct 27 '21 at 18:06
It works in this case, yes, @indmath. – Shaun Oct 27 '21 at 18:07
Yes, It will only work when we have even permutations and not in case of odd permutations. @Shaun – indmath Oct 27 '21 at 18:10
1

I believe so, @indmath. – Shaun Oct 27 '21 at 18:12

Breaking a cycle into 2 non-disjoint cycles

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