Sylow Theorems

A collection of three fundamental results about p-subgroups of a finite group: existence (there is a subgroup of order the maximal power of a prime p dividing the group order), conjugacy (all such maximal p-subgroups are conjugate), and counting (the number of these subgroups satisfies congruence and divisibility constraints).

A collection of theorems describing, for a finite group and a prime p dividing its order, the existence of p-subgroups of maximal p-power order (Sylow p-subgroups), their conjugacy properties, and congruence/counting constraints on their number.

A trio of fundamental results about p-subgroups in finite groups: existence (for a prime p dividing the group order there exists a subgroup of order p^n where p^n is the maximal p-power dividing |G|), conjugacy (all Sylow p-subgroups are conjugate), and counting (the number n_p of Sylow p-subgroups satisfies n_p ≡ 1 (mod p) and divides the p′-part of |G|).