1

Work out the decomposition in disjoint cycles.

I am working on disjoint cycles. I sometimes get confuse, so can anyone please check my work.

A) $(13)(2345) $

I am starting from right to left So $2$ goes $3$, and $3$ goes to $1$. $(21)$

Then $3$ goes to $4$ since there is no $4$ on the left, then I will have $(13)(2345) = (21)(345)$

and for F) $(12)(13)(14) = (14)(32)$

Are my answer correct. Thank you

mathematics2x2life
  • 13,322
Naye
  • 23
  • 2

2 Answers2

1

As you say (and most authors say), from right to left:

$$(13)(2345) \longrightarrow\begin{cases}1\to 3\\3\to 4\\4\to5\\5\to2\\2\to3\to1\end{cases}\;\implies (13)(2345)=(13452)$$

DonAntonio
  • 211,718
  • 17
  • 136
  • 287
0

When working with cycles, you need to continue with the last value done and keep going until the cycle actually cycles back on itself. You can't assume that you end up with disjoint cycles.

For (a), you've got $(21)$ so far. Instead of jumping to $3$, you need to find out what happens to $1$. So $1\rightarrow 1\rightarrow 3$, so now the cycle is $(213)$. Now do $4$ and $5$ to get the final answer of $(21345) = (13452)$. Note that you should check $5$ actually goes to $2$ as a double-check that things have worked out correctly.

For (f), you've got the numbers in a correct order but you don't end up with disjoint cycles. It is just a single cycle.

John Habert
  • 4,001
  • Why is (21345) = (13452)? Is it because they are commute – Naye Feb 11 '14 at 04:17
  • It is just a reorder of the terms. As long as items continuing moving in the right order, you can rewrite it however you like. So $(21345) = (13452) = (34521) = (45213) = (52134)$. We generally right the smallest number in each cycle first. – John Habert Feb 11 '14 at 04:21