If G be a cyclic group of prime order p ,prove that non identity element of G is a generator of the group. Let , a be the generator of the group . Then o(a)=p ==>a^p = e, where e be the identity element . G={a,a^2,a^3,.....,a^p(=e)}.
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3Do you know Lagrange’s theorem? – egreg Mar 05 '17 at 11:53
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@egreg Yes, i know Lagrange's theorem. – Saroj Ghosh Mar 05 '17 at 11:56
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3So, consider $\langle a\rangle$, which is a subgroup of $G$ having more than one element; hence $|\langle a\rangle|$ is a divisor of $p$ and is greater than $1$. – egreg Mar 05 '17 at 12:03
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1@egreg thanks i understand. I give you another question please solve this: prove that in a cyclic group of even order, there is exactly one element of order 2. – Saroj Ghosh Mar 05 '17 at 12:24