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Thread: Group & Number Theory Question

  1. #1
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    Group & Number Theory Question

    From Pinter 'A Book on Abstract Algebra' 2nd Ed.

    Question E-2 from Ch. 10 on page 109

    Let a and b be elements of a group G. Let ord(a)=m and ord(b)=n; let lcm(m,n) denote least common multiple of m and n. Prove if m and n are relatively prime, then no power of a can be equal to any power of b (except for e(the identity)).

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  2. #2
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    Re: Group & Number Theory Question

    You need to know the order of any element in a group is the order of the subgroup <x>. So suppose $j$ and $k$ are such that $a^j=b^k$. Since $<a^j>\subseteq <a>$, Lagrange says the order of $a^j$ divides m. In the same way $<a^j>\subseteq <b>$ and so the order of $a^j$ divides n. Hence the order of $a^j$ is 1; i.e. $a^j=e$.
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