For a prime number , a Sylow p-subgroup (sometimes p-Sylow subgroup) of a finite group is a maximal -subgroup of , i.e., a subgroup of that is a p-group (meaning its cardinality is a power of or equivalently: For each group element, its order is some power of ) that is not a proper subgroup of any other -subgroup of . The set of all Sylow -subgroups for a given prime is sometimes written .
foR a prime numBer , A sYloW p-suBGrOUp (sOmetimES p-sYlOw suBgRouP) of a FinitE GrouP is a maXimaL -subgROUp of , i.e., a SUBgroup oF thAT IS a p-gRoup (meaning itS caRdinALity is A power oF or EquivaLeNtly: for each group eLemEnT, iTs ORDEr is sOme PoWer of ) that is not a propeR sUbgRoUP of any otheR -SubgROup OF . the set of all sylOW -suBgrouPs FOr a GIVEn PRImE iS sOmEtiMes WritTen .