Permutation representations of finite simple groups: Orbits of cyclic ...

Proof: Let G 1 , .., G t be a system of representative for the
conjugacy classes of subgroups of G. Let Ω i := G/G i be the
corresponding G-sets, obtained by right multiplication. Suppose
Ω = ∑ n i Ω i and Ω ′ = ∑ n ′ i Ω i
are G-sets with fix Ω (U) = fix Ω ′(U) for all subgroups U in G.
Let J be the set of all i for which n i ≠ n ′ i
. Select j ∈ J so that
G j is maximal (wrt subconjugacy) among all G i with i ∈ J. Then
0 = fix Ω (G j ) − fix Ω ′(G j ) = ∑ i∈J
(n i − n ′ i ) · fix Ω i
(G j ).
Now fix Ωi (G j ) ≠ 0 only if G j is subconjugate to G i . So, only if
i = j by maximality. Hence
0 = ∑ i∈J
(n i − n ′ i ) · fix Ω i
(G j ) = (n j − n ′ j ) · fix Ω j
(G j ) > 0
which means that J = ∅.