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matt grime
Nov16-07, 02:59 AM
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P: 9,396
Quote Quote by JasonRox View Post
The textbook originally said G acts on X by conjugation. That was a typo for sure.
G does act on the set X by conjugation, since that is how it was defined. So does any subgroup of G, including H.

I don't really find all the details self-explanatory.
You don't find what part about your question self explanatory?

1. Let X be the set of G-conjugates of H, let [e],[g_1],..,[g_r] be a complete set of representatives of the conjugates (i.e. [g] is the conjugate gHg^-1, and inparticular [e] stands for the conjuate H of H).

2. H acts on X.

3. H sends the element [e] in X to [e], so there is at least one fixed point.

4. By the class equation for the action, this implies that there are at least p-1 fixed points.