crimboi[16] 搞笑群论

题目背景

人见人爱的代数小题 :] 惊人的是竟然在Hungerford II.2.8.9才出现, 还标了名字 😮

题面

A^ * is a normal subgroup of A, B^ * is a normal subgroup of B. Prove that:

(i) A^ *(A\cap B^ *) is a normal subgroup of A^ *(A\cap B)

(ii) B^ *(A^ *\cap B) is a normal subgroup of B^ *(A\cap B)

(iii) A^ *(A\cap B)/A^ *(A\cap B^ *)\cong B^ *(A\cap B)/B^ *(A^ *\cap B)

先想再看提示哦

提示

Actually (i)(ii)(iii) can be proved together by constructing an epimorphism f:A^ *(A\cap B)(B^ *(A\cap B)\text{ resp.})\rightarrow (A\cap B)/D with kernel A^ *(A\cap B^ *)(B^ *(A^ *\cap B)\text{ resp.})for some mysterious D :i (it's easy to guess using symmetry)

愿你无需看标答

标答

给了提示应该很简单了, 注意不要(也不需要)乱用AC!