Abstract algebra question concerning center of a group

[xcr]

Homework Statement

If a is the only element of order 2 in a group G, prove that a is an element of Z(G).
[Z(G) is the notation used by the book for center of group G]

Homework Equations

Z(G)={a is an element of G: ag=ga for every g that is an element of G}

The Attempt at a Solution

I know that if a has order 2 (|a|=2) then a ≠ the identity of the group, say e, and a=a^-1.
I just don't see where I would go from here in showing the center of a group.

 

[micromass]

What is the order of gag^{-1}??

 

[xcr]

I would say two but I don't really have any reasoning for saying that...

 

[xcr]

Actually, after looking at it, I would say that the order of gag^-1 is 2 because if the order of a is 2, then (a^2)=e. So (gag^-1)^2=(g^2)(a^2)(g^-2)=(g^2)(e)(g^-2)=(g^2)(g^-2)=e

 

[micromass]

Indeed. Now use that there is only one element of order 2...

 

[xcr]

Still don't see where you are going with it

 

[micromass]

There is only one element of order 2. What can you conclude??

 

[xcr]

That the element is not the identity and it is also its inverse.

 

[micromass]

You have found that both a and g^{-1}ag are elements of order 2.

But the question states that there is ONLY ONE element of order 2. So what can you conclude??

 

[xcr]

Then a=gag^-1. So multiplying on the right by g would give me ag=ga, ta-da