課程名稱︰代數導論優一 課程性質︰數學系必修 課程教師︰林惠雯 開課學院：理學院 開課系所︰數學系 考試日期︰2011年09月29日 考試時限：30分鐘(憑印象) 是否需發放獎勵金：是 試題： Honors Algebra (2011 Fall semester) Quiz One September 29, 2011 DERPARTMENT:＿＿＿＿＿＿　NAME:＿＿＿＿＿＿　ID NUMBER:＿＿＿＿＿＿ 1. (20%) Let S_n be the symmetric group of order n and σ∈S_n into 　　tranpostions. Prove that then number of factors occurring all have the 　　same parity, although the decomposition is not unique. 2. Choose exactly one problem form below. 　(a) (20%) Suppose a finite set G is closed under an associative product and 　　　that both cancellation laws hold in G, i.e., for any a, x, and y ∈G, 　　　ax=ay　→　x=y, and xa = ya →　x=y. 　　　Prove that G must be a group. 　(b) (20%) Let p be a prime and Z_p :={1,2,...,p-1}. For a,b∈Z_p, 　　　we define a＊b to be the integer r∈Z_p such that ab ≡ r (mod p). 　　　Show that (Z_p,＊,1) is a group. And then deduce the Wilson's theorem: 　　　(p-1)! ≡ (-1) (mod p) 　(c) (20%) Let G be a finite group of order n and g∈G. By Cayley's theorem, 　　　there exists an injective homomorphism ψ:G → S_n. Show that for g≠e, 　　　ψ(g) is a product of s disjoint t-cycles with st = |G|, i.e., 　　　　　ψ(g) = (a11 a12 ... a1t) (a21 a22 ... a2t) ... (as1 as2 ... ast), 　　　where aij∈{1,2,...,n}. And deduce that o(g)| |G|. --

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