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# Question on conjugacy classes

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1. Question on conjugacy classes
Prove that distinct conjugacy classes form a partition of a group G.

I don't really understand how to go about solving this one, does the question mean that I need to prove that a partition is formed when for all g in G, [g]={y in G, y=x-1gx for some x in G}?

thanks in advance for any help
Last edited by Lewk; 18-04-2012 at 22:52.
2. Re: Question on conjugacy classes
The general definition of a partition is as follows: let be a set. A collection of subsets of is a partition of if and if then .

That is, you split your set up into non-overlapping subsets.

With conjugacy classes, this means that you need to check that:
(i) every element is in some conjugacy class; and
(ii) if then
3. Re: Question on conjugacy classes
If you're familiar with them (and won't need to explicitly prove they have the properties they do) you could demonstrate 'is conjugate to' is an equivalence relation (thus forming a partition of the group with each element in an equivalence class and equivalence classes being equal or disjoint)

It won't save you any time here either way but it's a nice result (and can be used on relations outside group theory)

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Last updated: April 19, 2012
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