[試題] 105上 林惠雯 代數一 期末考
課程名稱︰代數一
課程性質︰數學系選修 可抵必修代數導論一
課程教師︰林惠雯教授
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰106.1.9
考試時限（分鐘）：3 hours
試題 :
1. (15%) Let M be any abelian group.
(a) Show that Hom (Z, M) is isomorphic to M.
Z
(b) Let n be a positive integer and M = {x in M|nx = 0}. Show that
n
Hom (Z/nZ, M) is isomorphic to M .
Z n
(c) Determine Hom (Z/nZ, Z/mZ) for n,m in N.
Z
2. (15%) Let 1 → N → E → G → 1 be an extension of an abelian group N by a
2
group G. Show that if H (G, N) = 0, then E is a semidirect product of N and G.
3. (15%) Show that the number of irreducible characters of a finite group G is
equal to the number of distinct conjugacy classes of G.
4. (20%)
(a) Show that every irreducible representation of a finite group G is
contained in the regular representation of G with multiplicity equal to its
degree.
(b) Show that
|K|χ(g')
Σ ρ(g) = --------- Id
g in K χ(1)
where ρ is an irreducible representation of a finite group G with character
χ, and K is a conjugacy class of G and g' is in K.
5. (25%) Determine the character tables of S and D . (Justify your answer)
4 6
6. (15%) Let a group G act on Z trivially and A be a Z[G]-module. Define the
augmentation map ε: Z[G] → Z by ε(Σm x) = Σm . Let I be the kernel of ε.
x x
Show that Z(tensor_Z[G])A is isomorphic to A/IA.
7. (10~30%) In case you are not confident that you can get over 60 points from
the above questions, state and show anything you have prepared.
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正妹也不過就是一組物質波方程式的特解罷了
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