The Non-communting Graphs Of Non-abelian Groups Of Orders10p~n

Let G be a finite group and let a non-commuting graph (?)(G) of G as follows: vertex set V (G)=G\Z(G) with two vertices x and y joined by an edge if and only if x and y is not the identity, denoted x-y. The number of edges incident with a vertex g is called the degree of g and denoted by deg(g). Clearly, it is easy to know the properties of(?)(G) if we know the structure of group G. But it is not easy to get G from (?)(G) For example (?)(D8)=(?)(Q8), but D8≠Q8.In this paper, we study the connection of the non-commuting graphs and the structure of groups of order10pn, where p>5. In Chapter2, we first classify the groups of order10pn with cyclic Sylow p-subgroup. And then we prove that for such group, the non-commuting graphs can decide their the groups (Theorem2.13). But if we give up the restriction on their Sylow p-subgroup, we find that the non-commuting graph can not decide the group. In order to show this, in Chapter3, we study the non-commuting graphs of the non-abelian groups of order10·72, and find two such groups with isomorphic non-commuting graphs (Theorem3.8).