 For a prime number "p", a Sylow "p"-subgroup (sometimes "p"-Sylow subgroup) of a group "G" is a maximal "p"-subgroup of "G", i.e., a subgroup of "G" that is a "p"-group (so that the order of every group element is a power of "p") that is not a proper subgroup of any other "p"-subgroup of "G".The set of all Sylow "p"-subgroups for a given prime "p" is sometimes written Syl"p"("G").The Sylow theorems assert a partial converse to Lagrange's theorem.