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So only recently encountering conjugation (in the group-theory sense) in my math adventures/education, and I can't help but ask why? It doesn't seem (at first glance) why its worthwhile defining such a term/homomorphism/idea. What do they really tell us about group structure? In $S_n$ they have the nice interpretation of equivalent cycle structures. For finite groups, conjugates can be thought of as having the same cycle structure in the encompassing symmetric group. But since a group on $n$ elements has far less than $n!$ elements, this interpretation isn't so useful.

Can someone offer an interpretation of what these equivalence classes are in a general group? Is the only reason to define them as such is so that we can define quotient groups?

Thanks for any help. Sorry if the post is broad/verbose/may not have an answer.

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    Thanks for the responses. Ya I can see the importance in being able to partition a group, and while it's true conjugate elements have the same order, not all elements of the same order are in the same conjugate class. What I'm trying to understand is; while I see that conjugation draws lines and partitions a group...I'm having a hard time seeing where and how these lines are drawn. If that makes any sense at all.. lol2012-11-12

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This is likely what youre looking for

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    precisely what I was looking for though lol. Thanks!2012-11-14