关于交换群的充分条件

On the Sufficient Conditions for the Abelian

利用指数及群同态的性质给出交换群的几个充分条件。证明了若G的指数为n,则G为交换群当且仅当n=2;若Ψ(x)=xn为群G上单同态映射,则当n=3,2,-1时或Φ(x)=xn-1是G上单映射或满映射时G为交换群。

In this paper ,some suffecient conditions for the abelian were obtained by utilizing the theory of exponent and the nomomorphis of group. It is domonstrated that when the exponent of group G is n, G is abelian if and only if n is 2. It is also shown that the group is the abelian if n is 3,2 or -1when ψ(x)=xn is the monomorphism on the group G.Similarly G is the abelian ifφ(x)=xn-1 is sinle_valued mapping or epimorphism.