The State Space Graph And Minimal Degree Of The Permutation Representation Of Small Order Groups

Finite groups are very important in group theory,whose structures and char-acters are widely used in many fields.But because of the highly abstract nature,it is needed to study some special groups first.Because the small order groups have become an important research object.In this paper,the graph and degree are used to study some small order groups.First,we characterize A5 by state space graph:Let G be a finite group with a homomorphism a.The vertex set of graph ?G,? is G;and for x,y are any two elements in G,there is an edge from x to y if and only if ?(x)= y.Such graph FG,?is called the state space of G about a.We discuss A5 with state graphs.The main results are as follows:Proposition 3.1 A5 can not be uniquely characterized by its state space which is induced by an isomorphism of order 2.proposition 3.2 A5 can not be uniquely characterized by its state space which is induced by an isomorphism of order 4.Theorem 3.3 Let G be a finite group,f be an automorphism of G,? be a 3 order isomorphism of A5.If ?G,f??A5,?,then G? A5.Theorem 3.4 Let G be a finite group,f be an automorphism of G,? be a 6 order isomorphism of A5.If ?G,f??A5,?,then G?A5.Then we mainly research the minimal degree of the permutation representation.If a positive integer d exists,d satisfies G(?)Sd and G(?)Sd-1 Then d is called the minimal degree of the permutation representation of G,it is d(G).we debate the minimal degree of the permutation representation of 56 and 60 order groups.Theorem 4.1 According to the classification of 56 order group[lemma 2.6].The minimal degree of the permutation representation of 56 order group.d(G1)= 15,d(G2)= 13,d(G3)= 13,d(G4)= 15,d(G5)= 11,d(G6)=15,d(G7)= 11,d(G8)= 15,d(G9)= 11,d(G10)= 7,d(G11)= 15,d(G12)=13,d(G13)= 8.Theorem 4.2 According to the classification of 60 order group[lemma 2.7].The minimal degree of the permutation representation of 60 order group.d(H1)= 12,d(H2)= 10,d(H3)= 10,d(H4)= 10,d(H5)= 9,d(H6)=8,d(H7)= 5,d(Hg)= 9,d(H9)= 10,d(H10)= 12,d(H11)= 9.