Let T be a property which holds for a group independent of whether or not this group is embedded in a group G or in a p-Sylow subgroup of G. Using a generalization of Sylow's second Theorem, we prove that if for any p-group P the number of subgroups of P satisfying T is congruent to 1 (mod p), then for any group G, the number of p-subgroups satisfying T is also congruent to 1 (mod p). As an application, we give simple proofs of several theorems, including the well-known Frobenius theorem.
Fulman, Jason and Vanderkam, Jeff, "When is the Number of p-Subgroups of a Group Satisfying a Property Congruent to 1 (mod p)?" (1993). Mathematical Sciences Technical Reports (MSTR). 85.