Wednesday, June 22, 2016

intuition - Interpretation of Group Conjugates



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 Sn 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.


Answer



This is likely what youre looking for


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