课程名称︰代数一 课程性质︰数学系选修 可抵必修代数导论一 课程教师︰林惠雯教授 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰105.11.14 考试时限（分钟）：3 hours 试题 : 1. (25%) (a) State and show the Chinese Remainder theorem. (b) State and show the Sylow(II) theorem. 2. (20%) (a) Show that no group of order 56 is simple. (b) Show that if |G| = 225, then there is a normal subgroup having prime index. 3. (25%) (a) Classify groups of order 8. (b) Classify groups of order 27 having a subgroup isomorphic to C . 9 4. (20%) Classify group of order 4p where p is an odd prime. 5. (10%) State the fundamental theorem of the finite abelian groups in the form of elementary divisor decomposition and show the uniqueness. 6. (25%) Let F be a finite field of order q > 5 and char(F) ≠ 2. G denotes a normal subgroup of SL(2, F) that is not contained in its center. (a) Compute |PSL(2, F)|. ╭1 λ╮ ╭1 0╮ (b) Show that SL(2, F) = 〈│ │,│ │|λ, λ' are in F〉. ╰0 1╯ ╰λ' 1╯ ╭0 -1╮ (c) Show that we may assume that G contains a matrix A = │ │. ╰1 c╯ ╭x y╮ (d) Show that G contains a matrix C = │ │ for some x ≠ ±1 and some y. ╰0 1/x╯ ╭1 λ╮ (e) Show that G contains all matrices │ │ with λ in F. ╰0 1╯ (f) Show that G = SL(2, F) and PSL(2, F) is a simple group. -- 正妹也不过就是一组物质波方程式的特解罢了 --

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