A New Characterization Of Several Classes Of Small Order Groups

Finite groups are very important in Group theory, whose structures and charac-ters are widely used in many relative subjects. But because of the abstract nature, it is needed to study some special groups first. The small order groups have become an important research object. In this paper, the graph and quantity are used to characterize some small order groups.First, we characterize groups by state space graph:Let G be a finite group with a homomorphism a. The vertex set of graph TG,α is G; and for x, y are any two elements in G, there is an edge from x to y if and only if α(x)= y. Such graph TG,α is called the state space of G about a. We discuss some small order groups with state graphs. The main results are as follows:Theorem 3.1 If ΓG,f ≌ΓA5,α, then G≌A5.Theorem 3.2 If ΓG,J ≌ΓS5,α, then G≌S5.Theorem 4.1The group defined by the special state space graph of the paper is D4 or the cyclic group of order 4. the special state space graph has only one cycle and p isolated vertexes.Then we characterize groups by quantity:Let G be a finite group. For X(?)G, We define φ(X)=Σx∈Xo(x),o(x) is the order of x. Let m(G) be the maximal order of elements in G, and We get:Theorem 5.2For a finite group G, then G≌S5 if and only if m(G)=6,5κ πe(G),φ(G)＝471.