Weak Isomorphism of Finite Group

On the basis of the quasi-isomorphism of finite groups, a new mapping, weak isomorphism, from a finite group to another finite group is defined. Let G and H be two finite groups and G be weak-isomorphic to H. Then G≌H if G satisfies one of the following conditions. 1） G is a finite Abelian group. 2） The order of G is p^3. 3 ） The order of G is p^n＋1 and G has a cyclic normal subgroup N = 〈a〉 of order p^n. 4） G is a nilpotent group and if p^││G│, then for any P ∈ Sylp （G）, P has a cyclic maximal subgroup, where p is a prime; 5） G is a maximal class group of order p4（p〉3）.