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If G is a group of a certain order, say 30, what are the possible sizes of conjugacy classes?

I know that I need to use the Class equation, but I'm not quite sure how to do it. Suggestions would be appreciated.

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    Are you asking for general notions, or are you asking about a group of order 30 specifically? (The "say 30" seems to imply that this is a number picked more or less at random). The order of a conjugacy class must divide the order of the group, and except for the trivial group, it cannot be equal to the whole group.2011-12-02
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    I'm asking about the order 30 specifically. In my notes, we were told that the order of the conjugacy class doesn't have to divide the order of the group, but the order of centralizers has to divide the order of the group. Was I mistold that?2011-12-02
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    For groups of size 30, the possible class sizes are 1, 2, 3, 5, 15, but of course one cannot have all of these sizes inside a single group. You can prove this using the class equation. Possible sizes within a single group of size 30 are {1}, {1,2,3}, {1,2,5}, and {1,2,15}. This seems to require a little more, but only sylow theorems really.2011-12-02
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    Thanks, Jack. Can you show how you obtained those sizes?2011-12-02
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    @user711: If the class equation tells you that the order of a centralizer times the order of a conjugacy class gives the order of the group, doesn't that tell you that both factors on the left divide the order of the group?2011-12-02
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    @JackSchmidt, how does the class equation rule out, say, $30=1+2+2+10+15$? How does it rule out classes of size 10, and classes of size 6?2011-12-03
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    @GerryMyerson: I answered in an answer since there are several cases to rule out. I was planning on saying the "not 6" in a comment, and then that "not 10 was similar, but a little harder", but I ran out of time even to comment. :-)2011-12-03

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