Elementary cyclic normal group theory

[eileen6a]

Homework Statement

If G is a finite group and let H be a normal subgroup of G with finite index m=[G]. Show that a^m\in H for all a\in G.

Homework Equations

order of a group equal the order of element.

The Attempt at a Solution

no idea.

 

[jbunniii]

Consider the canonical map \phi : G \rightarrow G/H defined by g \mapsto gH. What can you say about \phi(a^m)?

P.S. "order of a group equal the order of element" is false unless the group is cyclic and the element is a generator of the group.