How to determine if $\{n \cdot g^a \mod P\}$ and $\{m \cdot g^a \mod P\}$ generate the same sets? (set size < $P-1$)

Question

given some examples $k_{n_i},k_{m_i}$ out of each value set:
$k_{n_i} \in \{n \cdot g^a \mod P\, \forall a \in \mathbb{N}<P\}=G_m$
$k_{m_i} \in \{m \cdot g^a \mod P, \forall a \in \mathbb{N}<P\}=G_n$

Each set has size of $S$ which is a prime and known. Value $P$ is also a prime with $P = 2 \cdot S \cdot f+1$. Factor $f$ is (product of) prime(s) which is known as well. The Generator $g$ is known too. For a given $k$ the factors $n,m$ and related exponent $a$ is unknown.

As shown here for each $k$ multiple value pairs $(n,a)$ can be computed very fast (pick an $a$ and compute $n=kg^{-a} \mod P$). That means those sets can be equal with $n\not=m$.
Now is there a way to check if they generate the same sets (without computing all combinations?)

Answer 1

$G_n = G_m$ iff $n^S \equiv m^S \pmod P$

Proof:

If $n^S \not\equiv m^S \pmod P$, then $\forall e \in G_n : e^S = n^S$ (as $e^S = n^S \cdot (g^a)^s = n^S$); and similarly $\forall f \in G_m : f^S = m^S$. Hence $\forall e \in G_n, f \in G_m: e \ne f$, and hence $G_n \ne G_m$ (and actually the two sets are disjoint).

Other direction (needed because we're asserting equivalence):

If $n^S \equiv m^S \pmod P$, then $(nm^{-1})^S = 1$, that is $nm^{-1}$ is in the subgroup generated by $g$, that is, $g^c = nm^{-1}$ for some integer $c$.

Then, for any member $e \in G_n$, we have $e = n \cdot g^a$ (for some $a$); we have $n \cdot g^a = n \cdot g^{-c} \cdot g^{a+c} = n \cdot n^{-1}m \cdot g^{a+c} = m \cdot g^{a+c}$, and hence $e \in G_m$. Similarly, we can show that all elements $f \in G_m$ are also in $G_n$ and hence $G_n = G_m$

Extra credit for the reader: find the step where I implicitly assumed that $P$ was prime...