課程名稱︰代數導論一
課程性質︰系定必修
課程教師︰黃漢水 教授
開課系所︰數學系
考試時間︰2006/10/30 10:20-11:20
是否需發放獎勵金：否
試題 :
一 Let φ:Z24→S8 be group homomorphism such that
φ(1)=(23)(1584)
(1) Find φ(3),φ(4). (20%)
(2) Find yhe Ker(φ). (15%)
┌2┐
二 Let P=└4┘ be a point in the plane R^2
and φ:R^2→R^2 be a rotation about the point P through π/4.
┌a c┐ ┌p┐
Find an orthogonal matrix A=└b d┘ and v=└q┘ such that
┌x┐
for any u=└y┘ ∈R^2 φ(u)=Au+v
(30%)
三 Let Z21={0,1,2.....,20} and G={x∈Z21│(x,21)=1}.
We know that G is an abelian group with respect to multiplication.
(1)Prove that│G│=12 (10%)
(2)Find a,b,c∈G such that ord(a)=2, ord(b)=2, ord(c)=3
and G=〈a〉X〈b〉X〈c〉.(25%)
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