find the order of the elements in the group <Z*_{21}, .>
i think number of element in it is 20 from {1,2,3...20}
is it the divisors of 21 or divisors of 20?
please give an idea.
thank you.
The order of an element $\displaystyle o(a) = n$ is the least positive integer $\displaystyle n$ such that $\displaystyle a^{n} = 1_{G}$. In the case of $\displaystyle \mathbb{Z}_{21}^{*}$, we have $\displaystyle a^{n} \equiv 1 \pmod{21}$. So with $\displaystyle 1$, we clearly have $\displaystyle o(1) = 1$. With $\displaystyle 2$, we want $\displaystyle 2^{n} \equiv 1 \pmod{21}$. So ask yourself- $\displaystyle 2^{k} * 2 \equiv 1 \pmod{21}$. Solving for $\displaystyle 2^{-1} \equiv 32 \equiv 11 \pmod{21}$, and $\displaystyle 32 = 2^{5}$. So we get $\displaystyle 2^{6} = 64 \equiv 1 \pmod{21}$.
So does that help give you some intuition in solving for the orders of the other elements?