# Order of quotient group divides order of group

This fact is an application of the following pivotal fact/result/idea: Lagrange's theorem
View other applications of Lagrange's theorem OR Read a survey article on applying Lagrange's theorem
This article states a result of the form that one natural number divides another. Specifically, the (order) of a/an/the (quotient group) divides the (order) of a/an/the (group).
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## Statement

### Statement in terms of quotient groups

Let $G$ be a finite group and $N$ be a normal subgroup. The order of the quotient group $G/N$ divides the order of the group $G$.

### Statement in terms of surjective homomorphisms

Let $G$ be a finite group and $\varphi:G \to H$ be a homomorphism of groups. Then, the order of the subgroup $\varphi(G)$ of $H$ divides the order of $G$.

## Proof

### Proof of the statement in terms of quotient groups

Given: A group $G$, a normal subgroup $N$.

To prove: The order of $G/N$ divides the order of $G$.

Proof: By Lagrange's theorem (fact (1)), we have: $|G| = |N|[G/N|$.

This yields that the order of the quotient group $G/N$ divides the order of $G$.

### Proof of the statement in terms of homomorphisms

Given: A homomorphism of groups $\varphi:G \to H$, with $G$ a finite group.

To prove: The order of $\varphi(G)$ divides the order of $G$.

Proof: Let $N$ be the kernel of $\varphi$. By fact (2), $N$ is normal in $G$ and $\varphi(G) \cong G/N$. Since the order of $G/N$ divides the order of $G$, we obtain that the order of $\varphi(G)$ also divides the order of $G$.