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Math Help - If a is an element of order m in a group G and a^(k) = e, prove that m divides k.

  1. #1
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    If a is an element of order m in a group G and a^(k) = e, prove that m divides k.

    If a is an element of order m in a group G and a^(k) = e, prove that m divides k.
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  2. #2
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    Do a long division on the exponent.
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    Quote Originally Posted by FancyMouse View Post
    Do a long division on the exponent.

    I don't understand what you mean. Can you explain?
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    Quote Originally Posted by rainyice View Post
    I don't understand what you mean. Can you explain?
    Let k=rm+q where r is an integer and 0<=q<m. What can you say about a^q?
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  5. #5
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    Quote Originally Posted by FancyMouse View Post
    Let k=rm+q where r is an integer and 0<=q<m. What can you say about a^q?
    it has infinite order?
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  6. #6
    MHF Contributor chiph588@'s Avatar
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     k=rm+q , where  q<m .

    So,  1=a^k=a^{rm+q}=(a^{m})^r\cdot a^q = a^q

    Since  q<m , what is the only such  q that gives  a^q = 1 ?
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