First notice that the sum of the size of the fixed sets Sg is equal to the sum of the size of the stabilizer groups Gs: Both are counting the number of pairs (g,s) such that \(g\cdot s=s...First notice that the sum of the size of the fixed sets Sg is equal to the sum of the size of the stabilizer groups Gs: Both are counting the number of pairs (g,s) such that g⋅s=s. ∑g∈G|Sg|=∑s∈S|Gs|=∑s∈S|G|/|G⋅s|=|G|∑s∈S1|G⋅s|=|G|∑S/G1=|G||S/G|
