Permutation representations of finite simple groups: Orbits of cyclic ...
Permutation representations of finite simple groups: Orbits of cyclic ...
Permutation representations of finite simple groups: Orbits of cyclic ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
[2] : Let (G, Ω) be an action <strong>of</strong> the group G on the set Ω. Denote<br />
F := C and consider the permutation module F Ω for F G.<br />
For <strong>cyclic</strong> <strong>groups</strong> the permutation module is crucial in detecting<br />
regular orbits.<br />
Lemma (Brauer, 1940): Two n × n permutation matrices are<br />
conjugate in GL(n, C) if and only if the corresponding permutations<br />
are similar (the same up to renaming) in Sym n .<br />
In other words: for <strong>cyclic</strong> C we have F Ω ∼ = F ∆ as F C -modules<br />
if and only if (C, Ω) ∼ = (C, ∆) as G-sets. This lemma turns out<br />
to be very useful for our purposes.
Change language