Determine order of generator in GF(p)

Question

Given is prime $p$ and a generator value $g$ with $1 < g < p$. How can we efficiently compute the order of the subgroup that is created by

$g^k \mod p$

I've looked into Eulers Totient function, but can not seem to find an efficient way to compute this with no additional knowledge apart from $p, g$. Am I missing something or can this not be calculated efficiently?

Answer 1

If $g$ is a generator, the multiplicative group generated by $g$ is a cyclic group of order $p-1$. If $\gcd(p-1,k) = r$, the group generated by $g^k$ has order $(p-1)/r$: namely, it consists of $g^j$ for all $j$ divisible by $r$.