The Lagrange Theorem for groups shows that the order of a subgroup in a group has to be a factor of the order of the group. There is the inverse question of whether a group whose order has a factor k has a subgroup of order k. The Sylow (see loff) Theorems address this inverse question.

• First Sylow Theorem: If G is a group of order pⁿm with p a prime, n≥1 and p and m having no factor in common, then G contains a subgroup of order pⁱ for each value of i from 1 through n. Furthermore every subgroup of order pⁱ is normal in some subgroup of order pⁱ⁺¹ for 1≤i<n.

For a prime number p, a Sylow p-subgroup is a subgroup of order p that is maximal; i.e., it is not contained in any other subgroup of order p.

The conjugate of an element g of a group G with respect to an element c is cgc^-1. If H is a subgroup of G then the set of conjugate elements of H with respect to an element c, denoted as cHc^-1 is a subgroup of G (cHc^-1 may be H itself.)

• Second Sylow Theorem: If G is a group of order pⁿm with p a prime, n≥1 and p and m having no factor in common, and if H is a p-subgroup of G and S is a Sylow p-subgroup, then there exists an element g of G such that H is a subgroup of the conjugate of S with respect to g (gSg^-1).
   
  Corollary: Any two Sylow p-subgroups of G are conjugate.
• Third Sylow Theorem: If G is a group of order pⁿm with p a prime, n≥1 and p and m having no factor in common, then the number of Sylow p-subgroups is of the form kp+1 for some k≥0 and is a factor of the order of G.