Does Lagrange's Theorem Prove Every Element's Power Equals Group's Order?

[eileen6a]

Homework Statement

If G is a finite group with m elements. Show that $a^m=1$ for all a$\in$ G.

Homework Equations

order of group equal order of elements.

The Attempt at a Solution

i know order of group equal order of elements, but how to give a detailed proof?
Is G cyclic?

 

[Office_Shredder]

i know order of group equal order of elements

This isn't true. For example in Z₄, 2 has order 2, not 4. There is a statement you can make relating the order of a group element to the size of the group though

 

[eileen6a]

the question is wrong?but this is a question and i am supposed to prove it...

This isn't true. For example in Z₄, 2 has order 2, not 4. There is a statement you can make relating the order of a group element to the size of the group though

what is that statement??lagrange theorem?

 

[Mark44]

This isn't true.
For example in Z₄, 2 has order 2, not 4.

2 has order 2, so 2² = 1, but isn't 2⁴ also = 1?

There is a statement you can make relating the order of a group element to the size of the group though

 

[Office_Shredder]

a^m=1 does not mean that m is the order of a. The order of a is the smallest power you can raise a to that gives you one. So the question isn't asking you to prove that the order of a is the size of the group.

Yes eileena, Lagrange's theorem is what you need