刘觉平群论第五次作业.docVIP

Fill your name and your registration number in the above blanks! Please use the 2003 Word, Times New Roman for English letters, or 宋体（五号）for Chinese, and mathtype for equations (use the size for letters of 10.5, as convention ) to write your homework, and give the electronic files of your homework in time to the corresponding person being responsible for evaluating. The deadline for sending your homework through email is Sunday 24:00 of the week when the homework is given. Exercise 5 in Group Theory Exercises 5-1. Prove that there exists a homomorphism of the group onto the factor group of as follows 证明：令 则有 ∴与同态 Exercises 5-2. Let a group G act on the set by conjugation Show that 证明：假设,则，使得 ∴ ∵ ∴ ∴ Exercises 5-3. Let be a group acts on an arbitrary set and , the set of the points fixed by . Suppose that act on by conjugation , and act on the set by . Prove that the mapping defined by is a G-morphism. 解： ∴是 G-morphism. Exercises 5-4. Let a group G be the group of symmetries of the equilateral triangle; let be the set of three vertexes of the triangle. (1) Find ; (2) The mappingsandas and or simply expressed as,: Find , , , , , , , , . 解：（1）， ∴ （2） 与不存在。