# Homomorphism and periods

• Feb 16th 2011, 06:57 AM
Zalren
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.
• Feb 16th 2011, 07:17 AM
roninpro
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. 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, 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 $k=1, 2, 5,\text{ or } 10$.