Let p be a prime. Then, we know that $U(\mathbb{F}_p) \simeq \mathbb{Z}/(p-1)\mathbb{Z}$, where $U(\mathbb{F}_p)$ is the group of units of the field $\mathbb{F}_p$. They both have order $p-1$.
Given an integer $a$ , does it hold that:
$\bar a \in \mathbb{Z}/(p-1)\mathbb{Z}$ has order $p-1$ in the multiplicative group $U(\mathbb{F}_p)$
IF AND ONLY IF
$\bar a \in \mathbb{Z}/(p-2)\mathbb{Z}$ has order $p-1$ in the additive group $\mathbb{Z}/(p-1)\mathbb{Z}$ ?
They seem unrelated, but using $ (a,p) = (2, 11)$ and $ (a,p) = (2,17) $, I found that the two clauses are both true in both cases, so I was wondering if these two clauses have a relationship. What relationship do these two clauses have?
Thank you!
HINT
$\bar{1}$ has additive order $4$ and multiplicative order $1$ in $\mathbb{Z}/4\mathbb{Z}=U(F_5)$