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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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Personally, I think your question is just fine! –  amWhy Nov 12 '12 at 23:04
Well, equivalence relations are important in sets as they allows us to "divide" it in equivalence classes. As a rather important example is the equality relation, In case of conjugation in groups we have the above and also the important fact that conjugate elements have the same order. Many other things can be said, but the above is a good introduction, imo. –  DonAntonio Nov 12 '12 at 23:06
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.. lol –  user45793 Nov 12 '12 at 23:14
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1 Answer

up vote 1 down vote accepted

This is likely what youre looking for

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@Carl: Thomas didn’t say so, but the link takes you to an earlier question asking for the intuition behind conjugation; it isn’t quite the same as your question, but I agree that you may find the answers helpful. –  Brian M. Scott Nov 14 '12 at 5:59
A bit of elaboration on what is contained in the link would make a better answer. Links go dead, so it is better to give more than a link as an answer. –  robjohn Nov 14 '12 at 10:10
precisely what I was looking for though lol. Thanks! –  user45793 Nov 14 '12 at 15:31
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