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Math Help - Show that a nonabelian group must have at least five distinct elements.

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    Show that a nonabelian group must have at least five distinct elements.

    I have shown that there are distinct elements a and b st ab\neq ba. I have also shown that there is a unique identity element.

    Since G is a group, there a and b must have inverses. I have shown that a^{-1}\neq e, b^{-1}\neq e, a^{-1} \neq b and b^{-1}\neq a. What I can not find out is how to show that a^{-1}\neq a and b^{-1}\neq b. Does anyone have any ideas?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by auitto View Post
    I have shown that there are distinct elements a and b st ab\neq ba. I have also shown that there is a unique identity element.

    Since G is a group, there a and b must have inverses. I have shown that a^{-1}\neq e, b^{-1}\neq e, a^{-1} \neq b and b^{-1}\neq a. What I can not find out is how to show that a^{-1}\neq a and b^{-1}\neq b. Does anyone have any ideas?
    I am unsure what you are trying to prove-it seems to me that you are trying to prove that there exists a non-abelian group with 5 elements (which, curiously, there doesn't).

    I feel it is probably easiest to attack the problem by looking at the possibilities for groups of order 1, 2, 3 and 4. Try looking at their Cayley tables.
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    The question is asking me to prove that a non-abelian group has at least five distinct elements. But you're right I guess that I was trying to show that there was one with 5 elements, which is probably why I wasn't getting anywhere. Thanks!
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    Quote Originally Posted by auitto View Post
    I have shown that there are distinct elements a and b st ab\neq ba. I have also shown that there is a unique identity element.

    Since G is a group, there a and b must have inverses. I have shown that a^{-1}\neq e, b^{-1}\neq e, a^{-1} \neq b and b^{-1}\neq a. What I can not find out is how to show that a^{-1}\neq a and b^{-1}\neq b. Does anyone have any ideas?
    Instead of looking at a^{-1} and b^{-1}, try taking e,\ a,\ b,\ ab,\ ba as your five elements, and show that they are all distinct.
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    Thanks, that worked beautifully!
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