有限群的换位子群的分解

Decomposition of Commutator Subgroup of Finite Group

设G为有限群,A和B为G的子群,在G=AB且[A,B]=1条件下,利用换位子群的相关性质证明了G的换位子群G′=A′B,进一步,如果有A∩B=1,则G′∩A=A,G′∩B=B,,并利用矩阵构造群进行了结论的验证。

Let G be a fnite group,and A,B be its subgroups.It is proved that the commutator subgroup of G satisfies G'=A'B'under the condition ofG=A B and[A,B]=1.Furthermore,if A∩B=1,it is concluded thatG′=A′B,G′∩B=B′At last,the conclusion is validated through a group composed of two matrices.