Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b) $U$ is a domain of holomorphy and (c) $U$ is holomorphically convex. On the other hand, when $f , (, =(f_1,f_2,cdots,f_n), )$ is a $mathbb C^n$-valued function on an open set $U$ of $mathbb C^{k_1}timesmathbb C^{k_2}timescdotstimesmathbb C^{k_n}$, $f$ is said to be $mathbb C^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i ot=j$ implies $frac{partial f_i}{partial z_j}=0$. Here, $(z_1,z_2,cdots,z_n) in mathbb C^{k_1}timesmathbb C^{k_2}timescdotstimesmathbb C^{k_n}$ holds. We note that a $mathbb C^n$-analytic mapping and a $mathbb C^n$-analytic manifold can be easily defined. In this paper, we show an analogue of Cartan-Thullen theorem for a $mathbb C^n$-analytic function. For $n=1$, it gives Cartan-Thullen theorem itself. Our proof is almost the same as Cartan-Thullen theorem. Thus, our generalization seems to be natural. On the other hand, our result is partial, because we do not answer the following question. That is, does a connected open $mathbb C^n$-holomorphically convex set $U$ exist such that $U$ is not the direct product of any holomorphically convex sets $U_1, U_2, cdots, U_{n-1}$ and $U_n$ ? As a corollary of our generalization, we give the following partial result. If $U$ is convex, then $U$ is the direct product of some holomorphically convex sets. Also, $f$ is said to be $mathbb C^n$-triangular, if $f$ is complex analytic and for any $i$ and $j$, $i<j$ implies $frac{partial f_i}{partial z_j}=0$. Kasuya suggested that a $mathbb C^n$-analytic manifold and a $mathbb C^n$-triangular manifold might, for example, be related to a holomorphic web and a holomorphic foliation.
Let $G$ be a nonabelian group, $Asubseteq G$ an abelian subgroup and $ngeqslant 2$ an integer. We say that $G$ has an $n$-abelian partition with respect to $A$, if there exists a partition of $G$ into $A$ and $n$ disjoint commuting subsets $A_1, A_2, ldots, A_n$ of $G$, such that $|A_i|>1$ for each $i=1, 2, ldots, n$. We first classify all nonabelian groups, up to isomorphism, which have an $n$-abelian partition for $n=2, 3$. Then, we provide some formulas concerning the number of spanning trees of commuting graphs associated with certain finite groups. Finally, we point out some ways to finding the number of spanning trees of the commuting graphs of some specific groups.