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Finding center of a group with one element of order 2

  1. Sep 1, 2009 #1
    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?
     
  2. jcsd
  3. Sep 1, 2009 #2
    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?
     
  4. Sep 1, 2009 #3

    jambaugh

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    Try taking the square of aca^{-1}.
     
  5. Sep 1, 2009 #4
    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.
     
  6. Sep 1, 2009 #5

    jambaugh

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    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 _____"
     
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