I'm reading some lecture notes and watching some videos, but the purpose of group conjugation was never really made explicit. After a good amount of time, this is what I've come up with, in the context of the symmetric group $S_n$.
Every element of $S_n$ can be decomposed into disjoint cycles. There are $n!$ distinct cycle sets, as $S_n$ has cardinality $n!$. However, some cycles are equivalent to others, up to relabeling. If I have a cycle set $\sigma$ and want to obtain another equivalent cycle set, I can get any element of $S_n$, say $\tau$, and perform the following group composition $$\sigma'=\tau^{-1} \sigma \tau$$ The resulting element $\sigma'$ will have the same cycle set as $\sigma$.
If my understanding is indeed correct (if not please let me know!), I have a few questions.
The total number of cycle sets (equivalent ones that are relabeled are considered distinct) in $S_n$ is $n!$. If we count the equivalent ones as a single cycle set, then how many do we have left over? Is there a theorem for this? I am guessing it is some fraction of $n!$.
Is it safe to think of conjugation in this way? Because in other groups, there may not necessarily be a cycle set anymore. So what would conjugation by an element $g$ mean then? I understand that in an abelian group, conjugation trivially yields the exact same element. What about non-abelian groups?
Gun to head, if asked to explain conjugation of an element $h$ in $g$, in 10 words or less, my answer would be
It generates all of the cousins of $h$.
Would this roughly capture the essence of conjugation?
