In reply to a question by K. Dennis, J. Neubueser wrote:
> for all sporadic groups. Can one perhaps use character theory also for
> verifying C_2 or C_3? I have no idea, so here is another question. If
> so, we have plenty of charactertables and tools for handling them in
I don't think it is that easy to use character theory to verify C_2 or
higher. Character theory gives results "up to conjugacy in G", which is
sufficient for C_1, but not for C_2. A character theoretic verification
of C_1 essentialy verifies that for given x_1 there is a conjugacy class
(z) in G such that x_1 is in (z^-1)(z). Clearly , C_1 follows from this.
For given x_1, x_2, there is hope to find a conjugacy class (z) such that
x_1 and x_2 both are in (z^-1)(z), but this doesn't prove C_2, because you
don't know if it is "the same z" you get this way for writing x_1 and x_2
Nevertheless, GAP has been a very useful experimental tool for finding the
conjugacy class (z) in my thesis, and for a verification of C_2 or higher,
character theory will show you which conjugacy classes may contain the
wanted element z and which can't.
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