Let and be conjugate subgroups of , that is, for some , . Equivalently, , which means there is some element of such that under the action of on , its stabilizer subgroup is , all of the group of the group action. Suppose is a -group with index with respect to non-divisible by . Then such a fully stabilized coset must exist by the following lemma.
If is a -group that acts on , then , where is the subset of of elements fully stabilized by .
Its proof rests on the use orbit stabilizer theorem to vanish out orbits that are multiples of .
This is the natural origin of the second Sylow theorem.