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If |G| = p^a m with p ∤ m, then G has a subgroup of order p^a
Sylow's First Theorem. Let be a finite group, and write , where is prime and . Then has a subgroup of order . Such a subgroup is called a Sylow -subgroup.
The case includes Cauchy's theorem. Together, the three Sylow theorems describe the maximal -subgroups of .
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
Cauchy's Theorem (Finite Groups). Let be a finite group, and let be a prime number such that . Then there exists an element with and ; equivalently, has a cyclic subgroup of order .
Fix a prime number . A -group is a group such that every element has order for some integer (depending on ). If is finite, this is equivalent to saying that the cardinality of is a power of , i.e. for some .