I have a prime number $p$ and another number that divides $p - 1$, let's call it $l ~ | ~ (p - 1)$. Define their product as $q = l \cdot p$.
Moreover, I have a set $S = \left\{ 1 + i l ~ | ~ i \in 0, ..., p - 1; i \neq \frac{p - 1}{l} \right\}$. I want to show that $(S, \cdot)$ is a multiplicative, cyclic subgroup of the multiplicative group of integers modulo $q$, $(\mathbb Z / q \mathbb Z)^\times$.
Example: Choose $p = 19$, $l = 9$, so $q = 19 \cdot 9 = 171$, then $S = \{1, 10, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163\}$ with multiplication modulo $q$ is a cyclic multiplicative subgroup of $(\mathbb Z / q \mathbb Z)^\times$ with generators $g \in \{ 10, 91, 109, 127, 136, 154 \}$.
Any hints are greatly appreciated! Showing that $S$ is a subset is fairly straightforward, but I'm struggling to show that it is closed under multiplication and cyclic.