Question

在线等急，大哥大姐们谢谢了，一道抽象代数的题？

Let 𝐽 and 𝐾 be normal subgroups of a group 𝐺 and suppose the intersection of 𝐽 and 𝐾 is
the trivial subgroup. Let 𝑘 ∈ 𝐾 . Show that for every 𝑗 ∈ 𝐽 , we have 𝑗𝑘 = 𝑘 𝑗 .

Answer 1

证明：由于J，K是G的正规子群，所以对任意j∈J，k∈K，有：kjk^(-1)∈J，j^(-1)kj∈K.注意观察，此时有：j^(-1)kjk^(-1)∈J∩K，而J∩K={e}为平凡子群，因而j^(-1)kjk^(-1)=e，对此式两边同时左乘j以及右乘k即得：kj=jk，原命题得证.