Subgroup

Lagrange’s theorem

Given a group and subgroup , the order of the subgroup divides the order of the group. group This is often stated as

where is the number of unique (left or right) cosets of , and is called the Lagrange index.

Proof

Let . Any element is contained at least in the coset . Since Cosets are either identical or disjoint, cosets form a Partition of . Since is finite there is a finite number of cosets in the partition . The number of elements in each coset is equal to . Therefore, .

Corollary

The order of an element divides the order of a finite group , since forms a Cyclic subgroup.

Consequences

• All prime-ordered groups are cyclic

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