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︰2006/10/16 10:20-11:20
是否需發放獎勵金：否
試題 :
一 Let α,β,γ∈S5 such that
α=┌1 2 3 4 5┐ β=┌1 2 3 4 5┐ γ=(2 5 3 4)
└2 3 1 5 4┘ └3 5 1 2 4┘
compute (α^(-1))γβ , βγ(β^(-1)) .
(30%)
二 Let G be a group and a∈G
prove that H={x∈G│a*x=x*a} is a subgroup of G.
(30%)
三
Let G be a group, H be a subgroup of G and
K1={aH│a∈G} K2={Ha│a∈G}
Suppose that │K1│=4 and K1={a1H,a2H,a3H,a4H},
i.e. (1)for any a∈G,there is 1≦i≦4
such that aH=aiH.
(2)for any 1≦i<j≦4,aiH≠ajH
prove that K2={Hb1,Hb2,Hb3,Hb4} where bi=(ai)^(-1)
(40%)
[Hint:aH=bH←→(a^(-1))b∈H and Ha=Hb ←→b(a^(-1))∈H]
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