課程名稱︰代數導論一
課程性質︰數學系大二必修
課程教師︰莊武諺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰103/01/09
考試時限（分鐘）：150
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(1) (15 points) Please state 3 parts of Sylow theorem. (In the rest of test you could apply Sylow theorem directly.)
(2) (15 points) Consider the group action of S3 on S3 itself by left multiplication. Re-label the elements of S3 by {_1 = (1)(2)(3), _2 = (1 2)(3), _3 = (1)(2 3), _4 = (1 3)(2), _5 = (1 2 3), _6 = (1 3 2)}. Recall there is a bijection between the set of G action on the set A and the set of group homomorphisms from G to Sym(A). This bijection gives us a homomorphism Ψ:G → Sym(A), where G = S3 and A = S3.
Please write down the images of _1,...,_6 under the homomorphism Ψ in the cycle decomposition form in Sym(A).
(3) (15 points) Please classify the group of order 14. (Semidirect Product Recongnition Theorem could be assumed. Moreover, a complete classification should include the expression of every group element by group generators and the relations among the generators.)
(4) (10 points) (Cauchy theorem) Prove that if a prime p divides the order of a finite group G, then there exists an element of order p in G.
(5) (10 points) Prove that if |G| = 132 then G is not simple.
(6) (15 points) Let H and K be groups and let ψ:K → Aut(H) be a homomorphism. Prove that the following are equivalent:
(a) the identity set map between H ※ K and H × K is a group homomorphism.
(b) ψ is the trivial homomorphism from K to Aut(H).
(7) (15 points) Let R be a ring with identity. Prove that R is a division ring if and only if the only left ideals of R are (0) and R.
(8) (15 points) Let R be a commutative ring. Prove that the ideal M of R is maximal if and only if the quotient ring R/M is a field.
Remark: There are 110 points totally.
註：S3表3個元素的對稱群；H ※ K表the semidirect product of H and K with respect to homomorphism ψ