推 Bourbaki :我在想老師的名字好像打錯了 02/08 20:22
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※ 編輯: t0444564 來自: 140.112.252.31 (02/08 20:58)
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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