Let $\mathbb {Z}_n{\rm [i]}$ be the ring of Gaussian integers modulo $n$. We construct for $\mathbb {Z}_n{\rm [i]}$ a cubic mapping graph $\Gamma (n)$ whose vertex set is all the elements of\/ $\mathbb {Z}_n{\rm [i]}$ and for which there is a directed edge from $a \in \mathbb {Z}_n{\rm [i]}$ to $b \in \mathbb {Z}_n{\rm [i]}$ if $ b = a^3$. This article investigates in detail the structure of $\Gamma (n)$. We give suffcient and necessary conditions for the existence of cycles with length $t$. The number of $t$-cycles in $\Gamma _1(n)$ is obtained and we also examine when a vertex lies on a $t$-cycle of $\Gamma _2(n)$, where $\Gamma _1(n)$ is induced by all the units of $\mathbb {Z}_n{\rm [i]}$ while $\Gamma _2(n)$ is induced by all the zero-divisors of $\mathbb {Z}_n{\rm [i]}$. In addition, formulas on the heights of components and vertices in $\Gamma (n)$ are presented.
The article studies the cubic mapping graph $\Gamma (n)$ of $\mathbb {Z}_n[{\rm i}]$, the ring of Gaussian integers modulo $n$. For each positive integer $n>1$, the number of fixed points and the in-degree of the elements $\overline 1$ and $\overline 0$ in $\Gamma (n)$ are found. Moreover, complete characterizations in terms of $n$ are given in which $\Gamma _{2}(n)$ is semiregular, where $\Gamma _{2}(n)$ is induced by all the zero-divisors of $\mathbb {Z}_n[{\rm i}]$.
For a finite commutative ring $R$ and a positive integer $k\geqslant 2$, we construct an iteration digraph $G(R, k)$ whose vertex set is $R$ and for which there is a directed edge from $a\in R$ to $b\in R$ if $b=a^k$. Let $R=R_1\oplus \ldots \oplus R_s$, where $s>1$ and $R_i$ is a finite commutative local ring for $i\in \{1, \ldots , s\}$. Let $N$ be a subset of $\{R_1, \dots , R_s\}$ (it is possible that $N$ is the empty set $\emptyset $). We define the fundamental constituents $G_N^*(R, k)$ of $G(R, k)$ induced by the vertices which are of the form $\{(a_1, \dots , a_s)\in R\colon a_i\in {\rm D}(R_i)$ if $R_i\in N$, otherwise $a_i\in {\rm U}(R_i), i=1,\ldots ,s\},$ where U$(R)$ denotes the unit group of $R$ and D$(R)$ denotes the zero-divisor set of $R$. We investigate the structure of $G_N^*(R, k)$ and state some conditions for the trees attached to cycle vertices in distinct fundamental constituents to be isomorphic.
Let $FG$ be a group algebra of a group $G$ over a field $F$ and ${\mathcal U}(FG)$ the unit group of $FG$. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group $G$ with order $21$ over any finite field of characteristic $3$ is established. We also characterize the structure of the unit group of $FA_4$ over any finite field of characteristic $3$ and the structure of the unit group of $FQ_{12}$ over any finite field of characteristic $2$, where $Q_{12}=\langle x, y; x^6=1, y^2=x^3, x^y=x^{-1} \rangle $.