Given a group G, non-abelian, with |G|= , show G has distinct conjugacy classes. Any help would be great -- I'm sure there's a useful theorem or fact I'm missing out on. thanks!
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Originally Posted by kimberu Given a group G, non-abelian, with |G|= , show G has distinct conjugacy classes. Any help would be great -- I'm sure there's a useful theorem or fact I'm missing out on. thanks! let be the center of and the centralizer of in see that and also for all because now apply the class equation.
So the class equation says: G=Z(G)+[G:C( )] meaning, this sum, but what exactly is the sum? Is it just p, m times? (sorry, I've never used this equation before. thanks so much for the help!)
Originally Posted by kimberu So the class equation says: G=Z(G)+[G:C( )] meaning, this sum, but what exactly is the sum? Is it just p, m times? (sorry, I've never used this equation before. thanks so much for the help!) the in your sum in the right hand side are not in and thus, as i mentioned, now suppose is the number of conjugacy classes. then: which will give us the result.
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