In this note we solve Noether's problem over $\mathbf{Q}$ for some meta-abelian groups of small degree $n$. Let $G$ be a subgroup of the group of one-dimensional affine transformations on $\mathbf{Z}/n\mathbf{Z}$ which contains $\mathbf{Z}/n\mathbf{Z}$. For $n=9,10,12,14,15$, we show that Noether's problem for $G$ has an affirmative answer by constructing an explicit transcendental basis of the fixed field over $\mathbf{Q}$.

Let $q > 2$ and $c$ are two integers with $(q, c) = 1$, for each integer $a$ with $0 < a < q$ and $(a, q) = 1$, there exists one and only one $b$ with $0 < b < q$ such that $ab \equiv c \pmod{q}$. Let \[ M(q,k,c,n) = \underset{a_1 \dotsm a_n b \equiv c \pmod{q}} {\sideset{}{'}\sum_{a_1=1}^q \dotsm \sideset{}{'}\sum_{a_n=1}^q \sideset{}{'}\sum_{b=1}^q} (a_1 \dotsm a_n - b)^{2k}, \] the main purpose of this paper is to study the asymptotic behavior of $M(q,k,c,n)$, and prove that for any positive integers $k$ and $n$ with $n \ge 2$ we have \[ M(q,k,c,n) = \frac{\phi^n(q) q^{2kn}}{(2k+1)^n} + O \Bigl( 4^k q^{(2k+1)n - (1/2)} d^2(q) \ln q \Bigr). \]

We give a characterization of the first standard imbedding of the Cayley projective plane into a real space form in terms of a particular class of Frenet curves of order 2.