3

Let for example $a=(14395)(26)(78)$ and $b = (154)(2368)(79)$ be elements of $S_9$.

I know that by definition, conjugate elements of a group $G$ are elements $x,y \in G$ such that $x=aya^{-1}$ for some $a \in G$.

The definition doesn't tell what it really represents and why it is important.

What is the idea behind conjugation and how can I tell if two cycles are conjugate?

E W H Lee
  • 2,336
khajvah
  • 837

1 Answers1

5

Compare the cycle structures, the permutations are conjugate if and only if they have have the same cycle structure. So the two exhibited above are not conjugate since one has structure 5,2,2 and the other has structure 4,3,2. For two which are conjugate, you get a conjugating permutation by writing the one permutation above the other, lining up the cycles of the same size (including the 1 cycles), and then mapping "down" from the top elements to the bottom ones (I.e. Conjugation is relabeling).

Ned
  • 3,852
  • I would like to note that cycle structure is only sufficient when you have elements that can do the actual conjugation in the group. In case of the permutation representation of the $Q_8$ group for example labelled in the order ${1,-1,i,-i,j,-j,k,-j}$, (1324)(5768) is the $i$ element, which is not conjugate with $j$, which is (1526)(3847). They would be in $S_8$ though. – Balint Pato Oct 04 '21 at 21:45