Show there exist an element of order 2 in this group

[halvizo1031]

Homework Statement

If G is a finite groups whose order is even, then there exists an element a in G whose order is 2.

Homework Equations

The Attempt at a Solution

does this mean that a^2 is the identity? how can i prove this? Also, would't this make G cyclic?

 

[VeeEight]

Yes, that would make a² = 1
Perhaps you should check out Cauchy's Theorem

 

[HallsofIvy]

Homework Statement

If G is a finite groups whose order is even, then there exists an element a in G whose order is 2/

Homework Equations

The Attempt at a Solution

does this mean that a^2 is the identity? how can i prove this? Also, would't this make G cyclic?

Yes, that means that a^2 is the identity. And that, in turn, means that a is its own inverse.

The group identity has the property that it is its own inverse (but has order 1, not 2). Suppose that there were no other member of the group with that property. Then we could pair the other members of the group, each paired with its inverse.