My Algebra Questions: Answers & Updates

[matt grime]

Assuming you mean the standard class equation (the sum of the orders of the conjugacy classes) then you need to write down what your index set is otherwise the question is meaningless.

Properly, the class equation really should sum over all conjugacy classes, so of course there will always be an i_j equal to |G| - the one for the identity element.

Of course some people concatenate all the elements in the centre into a single term, and say it is

|Z(G)| + sum over remaining conjugacy classes.

 

[matt grime]

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.

 

[JasonRox]

I meant details to proofs in the textbook. It definitely doesn't read like a novel sometimes.