Let be a homomorphism. Let be an element of of period 10. Prove that must have period 1 or 2 or 5 or 10.

I can prove that may have period 10.

because have period 10

by property of homomorphism

Therefore may have period 10.

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- February 16th 2011, 07:57 AMZalrenHomomorphism and periods
Let be a homomorphism. Let be an element of of period 10. Prove that must have period 1 or 2 or 5 or 10.

I can prove that may have period 10.

because have period 10

by property of homomorphism

Therefore may have period 10. - February 16th 2011, 08:17 AMroninpro
Suppose that has period . Then by the division algorithm, we can write where are integers and . Then, following your original calculation,

First note that as was the period. This means that . But since , we must take (i.e. was the smallest nonzero number such that ). This means that , so the period must divide 10. So we can conclude that .