Queston;
Given that 2 is a generator of cyclic group U(25), find all generators.
I am only conversant with the finding the mod which is very long with this question. Pls can someone enlighten me on how to get it done faster.
I am new to the cyclic group.
solution
U(25) = {1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19,21,22,23,24}
2^20 = 1 (mod 25)
Thanks
Best Answer
In a cyclic group of order $n$ generated by $g$, the order of $g^k$ is $\dfrac{n}{\gcd(n,k)}$.
In particular, the generators are $g^k$ with $\gcd(n,k)=1$.
In your case, $g=2$ and $n=\phi(25)=20$.
Therefore, the generators of $U(25)$ are $2^k$ for $k$ coprime with $20$, that is, $k$ odd not a multiple of $5$.