Lagrange's theorem

Given a group $𝐺$ and subgroup $𝐻 \subseteq 𝐺$ , the order of the subgroup divides the order of the group. #m/thm/group This is often stated as

| 𝐺 | = | 𝐺 / 𝐻 | | 𝐻 |

where $| 𝐺 / 𝐻 |$ is the number of unique (left or right) cosets of $𝐻$ , and is called the Lagrange index.

Proof

Let $𝐻 \subseteq 𝐺$ . Any element $𝑔 \in 𝐺$ is contained at least in the coset $𝑔 𝐻$ . Since Cosets are either identical or disjoint, cosets form a Partition of $𝐺$ . Since $𝐺$ is finite there is a finite number of cosets in the partition $(𝐺 : 𝐻)$ . The number of elements in each coset is equal to $| 𝐻 |$ . Therefore, $| 𝐺 | = (𝐺 : 𝐻) | 𝐻 |$ .

Corollary

The order $| 𝑔 |$ of an element $𝑔 \in 𝐺$ divides the order $| 𝐺 |$ of a finite group $𝐺$ , since $⟨ 𝑔 ⟩$ forms a Cyclic subgroup.

Consequences

All prime-ordered groups are cyclic

#state/tidy | #lang/en | #SemBr