## The graph of the square mapping on the prime fields.(English)Zbl 0843.05048

Let $$f: G\to G$$ be a map of a finite set into itself. Then $$\text{graph}(G)$$ is defined to be the directed graph whose vertices are the elements of $$G$$ with a directed edge from $$x$$ to $$f(x)$$ for all $$x\in G$$. In this paper, the action of the map $$f(x)= x^2$$ on the multiplicative groups $$F^*_p$$ of the prime fields is examined. Let $$C_m$$ denote the cyclic group of order $$m$$, and suppose that $$m$$ is odd. Then $$\text{graph}(C_m)$$ relative to $$f$$ is the disjoint union $\text{graph}(C_m)= \bigcup_{d|m} \undersetbrace {\varphi(d)/\text{ord}_d 2}\to {(\sigma(\text{ord}_d 2)\cup\cdots\cup \sigma(\text{ord}_d 2))},$ where $$\sigma(l)$$ is the cycle of length $$l$$, $$\varphi$$ is Euler’s function and $$\text{ord}_d2$$ is the order of $$2\text{ mod } d$$. Let $$n= 2^km$$ where $$m$$ is odd. Then $\text{graph}(C_m)= \bigcup_{d|m} \undersetbrace{\varphi(d)/\text{ord}_d 2}\to {(\sigma(\text{ord}_d 2, k)\cup\cdots \cup \sigma(\text{ord}_d 2, k))},$ where $$\sigma(l, k)$$ consists of a cycle of length $$l$$ with a copy of the binary tree $$T_k$$ of height $$k$$ attached to each vertex.

### MSC:

 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

### Keywords:

decomposition; cyclic group; cycle; tree
Full Text:

### References:

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