Order of quotient group divides order of group

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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

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

  1. Lagrange's theorem
  2. First isomorphism theorem

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.