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Exercise 2 in Group Theory90 Please give the electronic files (WORD2003 document ) of your homework in time to the corresponding person being responsible for evaluating. Deadline for sending your homework through email is Sunday 24:00 of the week when the homework is given. Problem 1. Show that a group must be an Abelian group if the order of any element in the group, except for the identity , is 2. 解：设，有，，故， 又由群的封闭性可知 ，则，即 两边左乘，右乘，得，化简得 故群为阿贝尔群，证毕。 Problem 2．(Lorentz group) Prove that in Minkowski space-time where the four coordinates of an event is expressed as , all the Lorentz tansformations along with the X-axis (namely all boosts along with X-axis) of the form where ,, form a group. 解：我们用矩阵来表示洛伦兹变换 ，记此矩阵为 对于任意的和， 其中，故封闭性满足 故，结合律满足 存在单位变换矩阵，使得 对于某一个元素矩阵，总是存在，使得 故洛伦兹变换矩阵组成一个群，证毕 Problem 3. Let G be a finite cyclic group, and let n be a positive integer which divides the order of G , . Prove that G has a cyclic subgroup of G of order n, . 解：G为有限循环群，设，则，故存在某元素，满足，则 令，则 故 故群为G的一个循环子群，证毕。 Problem 4. Let . Let x,y be the permutations in which are given by ,,and let K be the subgroup of . Show that the function , defined by (), is a homomorphism. (Notation： means the subgroup generated by the group elements x and y.) 解： 由，中的乘法表为 中的乘法表为 故映射 ()，即 由两乘法表可知 对于，， 有 其中与同奇偶性。故映射为的同态。证毕。 a与b不能交换位置，第一个等号不成立。 例如 扣10分。