Groups Of All Automorphisms And Holomorphs Of A Class Of Finite Groups

Let G be a finite group.G is presented with generators and defining relators which isFirst,this paper obtains a feasible method of calculating the order of automorphism group Aut(G) of finite groups G.that is,Then,we solve the order of automorphism groups of dihedral groups D₂m(m ≥ 3) and abelian p-groups of type [1, n].Second,making use of the knowledge that concerns the semidirect product of finite group and the theory of number ,solves the explicit structure of automorphism groups of D₂pⁿ (p ≥ 3) and D₂²pⁿ (p ≥ 3) ,furthermore discusses the form of construction of automorphism groups of D₂m(m ≥ 3).When n ≥ 3,discusses the explicit structure of groups of all automorphisms of the others 3 kinds of mutually nonisomorphic non-commutative groups where their common order is 2ⁿ⁺¹ and each kind of group has a cyclic normal subgroup N — (a) whose order is 2ⁿ. Solves the explicit structure of automorphism groups of abelian 3-groups of type [1, n] and the set of generators of abelian p(p ≥ 3) groups of type [1,1] .obtains a presentation of groups of all automorphisms of abelian P (p≥ 3)-groups of type [l,n] (n ≥ 2).Last,we discuss the holomorph of finite groups .