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.

skauvzc

skauvzc

Answered question

2022-09-17

Difference of fixed points by subgroup action
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 | + | 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.

Answer & Explanation

Santiago Collier

Santiago Collier

Beginner2022-09-18Added 8 answers

Step 1
The 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 2
in 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 .

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