  Let $\mathbb{A}[n]$ be the group of $n$-torsion points of a commutative
algebraic group $\mathbb{A}$ defined over a number field $F$. For a prime ideal
$\mathfrak{p}$, we let $N_{\mathfrak{p}}(\mathbb{A}[n])$ be the number of
$\mathbb{F}_\mathfrak{p}$-solutions of the system of polynomial equations
defining $\mathbb{A}[n]$ when reduced modulo $\mathfrak{p}$. Here,
$\mathbb{F}_{\mathfrak{p}}$ is the residue field at $\mathfrak{p}$. Let
$\pi_F(x)$ denote the number of primes $\mathfrak{p}$ of $F$ whose norm
$N(\mathfrak{p})$ do not exceed $x$. We then, for algebraic groups of dimension
one, compute the $k$-th moment limit $$M_k(\mathbb{A}/F, n)=\lim_{x\rightarrow
\infty} \frac{1}{\pi_F(x)} \sum_{N(\mathfrak{p}) \leq x}
N_{\mathfrak{p}}^k(\mathbb{A}[n])$$ by appealing to the prime number theorem
for arithmetic progressions and more generally the Chebotarev density theorem.
We further interpret this limit as the number of orbits of the action of the
absolute Galois group of $F$on $k$ copies of $\mathbb{A}[n]$ by an application
of Burnside's Lemma. These concrete examples suggest a possible approach for
determining the number of orbits of a group acting on $k$ copies of a set. We
also show that for an algebraic set $Y$ of dimension zero, the corresponding
arithmetic function $N_\mathfrak{p}(Y)$, defined on primes $\mathfrak{p}$ of
$F$, has an asymptotic limiting distribution.
