Difference of fixed points by subgroup actionIf G is a finite group and X is a finite G-set, then we have a class equation which tells us | X | = | X G | + ∑ | G / G x | where X G is the set of G-fixed points, G x is the stabilizer of x ∈ X and the sum is taken over representative elements of classes of non-singleton G-orbits.My question, which arise from a proof in a paper which seems to use precisely this fact, is the following. If H ≤ G is a normal subgroup, are we able to write the cartinality of X H ∖ X G as sum of cardinalities of G/H orbits? If this is the case, over what that sum is taken?I guess the result should descend by comparing the two class equations for H and G, but I'm struggling to write that down.

Question

Difference of fixed points by subgroup actionIf G is a finite group and X is a finite G-set, then we have a class equation which tells us      |    X      |    =      |        X    G        |    +  ∑      |    G      /        G    x        |  where       X    G   is the set of G-fixed points,       G    x   is the stabilizer of   x  ∈  X and the sum is taken over representative elements of classes of non-singleton G-orbits.My question, which arise from a proof in a paper which seems to use precisely this fact, is the following. If   H  ≤  G is a normal subgroup, are we able to write the cartinality of       X    H    ∖      X    G   as sum of cardinalities of G/H orbits? If this is the case, over what that sum is taken?I guess the result should descend by comparing the two class equations for H and G, but I&#039;m struggling to write that down.

Santiago Collier · Accepted Answer

Step 1The group G/H acts on       X    H    ∖      X    G   without fixed points, and so the equation you cited gives      |        X    H    ∖      X    G        |    =  ∑      |    (  G      /    H  )      /    (  G      /    H      )    x        |  Step 2in this case where the sum is over representative elements of the G/H-orbits (which are the same as the G-orbits) of       X    H    ∖      X    G  .

If G is a finite group and X is a finite G-set, then we have a class equation which tells us |X|=|X^G|+sum |G/G_x| where X^G is the set of G-fixed points, G_x is the stabilizer of x in X and the sum is taken over representative elements of classes of non-singleton G-orbits.

Answered question

Answer & Explanation

New Questions in College Statistics