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Math Help - Proves in Group Theory

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    Proves in Group Theory

    Let a and b be elements of a finite group G.
    a. Prove that a and a^(-1) have the same order.
    b. Prove that a and b*a*b^(-1) have the same order.
    c. Prove that a*b and b*a have the same order.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rainyice View Post
    Let a and b be elements of a finite group G.
    a. Prove that a and a^(-1) have the same order.
    b. Prove that a and b*a*b^(-1) have the same order.
    c. Prove that a*b and b*a have the same order.
    One word hind: Induction.
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    Quote Originally Posted by Swlabr View Post
    One word hind: Induction.

    Do you mean do each part of the question by induction?
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Swlabr View Post
    One word hind: Induction.
    You don't necessarily need induction for any of these (or so I think -- I know its overkill for (a)!). For question (a), just consider the two cases where a has infinite order and where a has finite order (|a|=n)
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    You don't necessarily need induction for any of these (or so I think -- I know its overkill for (a)!). For question (a), just consider the two cases where a has infinite order and where a has finite order (|a|=n)
    Sure, but it is the most obvious way and works for all parts of the question.
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  6. #6
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    Actually induction is not necessary. To show that "x" has the same order of "y", prove o(y)<=o(x) first, and then next o(x)<=o(y). Fortunately each one of the three cases is symmetric, i.e. you don't need to do anything to prove o(x)<=o(y) since you already got o(y)<=o(x)
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