[Lewk]

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

[nuodai]

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

[sputum]

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)