Finding center of a group with one element of order 2

[Sprooth]

I need to prove that if a group contains exactly one element with order 2, then that element is the center of the group.

Here is how I formulated the problem:
Let A be a group with an element c such that c^2 = 1 (i.e. c = c^-1), and b^2 = 1 implies b=c.

Want either:
ac = ca, for all a in A
ca(c^-1) = a, for all a in A.

After doing this, I tried manipulating the symbols for a while using different identities and properties that I know, but I can't seem to come up with any equations where one side doesn't reduce to the other trivially. I have a feeling that there is one small but crucial thing I am missing. Abelian groups have crossed my mind a few times, but I don't know how to use that.

Does anyone have any suggestions for where to go from here?

 

[fantispug]

b^2=1 actually implies b=c OR b=1.

So the information we have is:
b^2=1 iff b=c or b=1.

And we're trying to show:
ac = ca for all a in A.

So we need to somehow convert our problem into one where we can use ALL the information we have. So can you convert what we are trying to show into b^2=1 (keeping in mind c^2=1) for some b?

 

[jambaugh]

Try taking the square of aca^{-1}.

 

[Sprooth]

fantispug: Thanks for pointing out that b^2 = 1 implies either b = 1 or b = c. I hadn't thought of that. Taking that into consideration, I played around with the equations some more, but there was something that confused me a little bit. When considering b with b^2 = 1, when b is 1 or c must be treated as separate cases, right?

jambaugh: I think your hint may have given me the connection I needed, although it still feels like I'm not doing everything correctly.

So is this in the right direction of the proof?
1 = b^2
aa^{-1} = b^2
acca^{-1} = b^2
aca^{-1} * aca^{-1} = b^2
(aca^{-1})^2 = b^2 = c^2
(aca^{-1})^2 = c^2
aca^{-1} = c // I'm not sure about the step leading to this. Can you take roots like that?
ac = ca
So c is the center of the group.

Could someone please tell me if I skipped any steps or made any incorrect steps or if I am just not getting it? I'm just starting algebra this semester, and I think it's interesting, but I'm having a hard time getting into the right mindset so far.

Thanks for the help.

 

[jambaugh]

I think you have all the steps but you should work on the sequence of inferences. You should include some text especially when invoking the assumptions. I'm thinking something like:

"Observe that __=__."
"Thus by our assumption ___ either ___ or ___ ."
"If (case 1) then ...
"If (case 2) then ...
"Thus it is shown that _____"