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Math Help - Homomorphism and periods

  1. #1
    Junior Member
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    Homomorphism and periods

    Let f : G -> G' be a homomorphism. Let a be an element of G of period 10. Prove that f(a) must have period 1 or 2 or 5 or 10.

    I can prove that f(a) may have period 10.

    a^{10} = e because a have period 10
    f(a^{10}) = [f(a)]^{10} = e' by property of homomorphism
    Therefore f(a) may have period 10.
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  2. #2
    Senior Member roninpro's Avatar
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    Suppose that f(a) has period k. Then by the division algorithm, we can write 10=qk+r where q,r are integers and 0\leq r<k-1. Then, following your original calculation,

    e'=f(a^{10})=f(a^{qk+r})=f(a^{qk})f(a^r)=[f(a)]^{qk}[f(a)]^r

    First note that [f(a)]^{qk}=e' as k was the period. This means that [f(a)]^r=e'. But since r<k, we must take r=0 (i.e. k was the smallest nonzero number such that [f(a)]^k=e'). This means that 10=qk, so the period k must divide 10. So we can conclude that k=1, 2, 5,\text{ or } 10.
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