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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).
View other divisor relations |View congruence conditions

Contents

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.

Related facts

Closely related facts

Applications

Other facts about order dividing

Facts used

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.