[試題] 105上 林惠雯 代數一 期中考
課程名稱︰代數一
課程性質︰數學系選修 可抵必修代數導論一
課程教師︰林惠雯教授
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰105.11.14
考試時限（分鐘）：3 hours
試題 :
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.
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正妹也不過就是一組物質波方程式的特解罷了
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