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Math Help - If Ord(a)=1 iff a=e

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    If Ord(a)=1 iff a=e

    Let a,b,c be elements in G Prove the following.

    Ord(a)=1 iff a=e.


    Attempt: Could i just use definition of order of a group ? So if Ord(a)=1 then the size of G is a?
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  2. #2
    Junior Member beebe's Avatar
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    Re: If Ord(a)=1 iff a=e

    If a,b,c are distinct elements, then the order of G is 3 or greater. Use the definition of order for group elements. That is, o(x)=n for the smallest positive integer n which satisfies x^n=e.
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    Re: If Ord(a)=1 iff a=e

    if o(a) = 1, this means that a =a^1 = e. so that's the "if" part.

    suppose that a = e. then surely a^1 = a = e. so 1 is a positive integer k such that a^k = e. since there is no positive integer smaller than 1, 1 is clearly the least positive integer with this property: that is, o(a) = 1.

    to use this definition: o(a) = |<a>|, note that if |<a>| = 1, then <a> has just one element. this means that <a> contains the identity of G, since every subgroup of G contains the identity, and since a is the only element of <a>, a = e (if part). if a = e, then <a> = \{e^k : k \in \mathbb{Z}\}, and since e^k = e for all integers k, <a> = <e> has just one element (namely, a = e), so |<a>| = |<e>| = 1 (only if part).
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