課程名稱︰代數一
課程性質︰數學系大二必修
課程教師︰于靖
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/11/16
考試時限（分鐘）：180
試題 :
In answering the following problems, please give complete arguments as much as
possible. You may use freely any Theorem already proved (or Lemmas,
Propositions) from the Course Lectures, or previous courses on Linear Algebra.
You do not need to give proofs of the theorems you are using, but you MUST write
down complete statements of the theorems which your arguments are based.
1. Let F be a field. Let SL (F) be the special linear group consisting of n ×n-
n
matrices with entries from F and having determinant 1. Prove that the center
Z(SL (F)) of the group SL (F) is isomorphic to the multiplicative group
n n
×
consisting of n-th roots of 1 inside F . (Hint: use elementary matrices with
(i, j)-entry 1, when i ≠ j, and the other off diagonal entries 0.)
2. Let |F be the field with 3 elements. Let group G := PSL(2, |F ) := SL (|F )
3 3 2 3
/ {±I}. Prove that G \cong A , the alternating group of degree 4. (Hint:
4
╭ 0 2 ╮ ╭ 1 1 ╮ ╭ 1 1 ╮
consider the matrices │ │, │ │, │ │ modulo 3.)
╰ 1 0 ╯ ╰ 1 2 ╯ ╰ 0 1 ╯
3. Let G be a group. Define Z (G) := 1, Z (G) := Z(G) the center of G, and
0 1
inductively for i ≧ 1, Z (G) is the subgroup of G containing Z (G) such
i+1 i
that
Z (G) / Z (G) = Z(G/Z (G)).
i+1 i i
Prove that for all i, Z (G) is a characteristic subgroup of G. In other
i
words, any automorphism of G maps Z (G) to Z (G).
i i
4. (ⅰ) Show that the Sylow 3-subgroups of the symmetric group S are non-
9
abelian of order 81.
(ⅱ) Let Ｚ be the cyclic group of order 3, and φ: Ｚ → Aut(Ｚ ×Ｚ ×
3 3 3 3
Ｚ ) be the homomorphism sending the generator of Ｚ to the
3 3
automorphism of H := Ｚ ×Ｚ ×Ｚ which corresponds to the cycle
3 3 3
permutation (1 2 3) of the three factors of the direct product Ｚ ×Ｚ
3 3
×Ｚ :
3
_
φ(1): (x, y, z)├→ (y, z, x).
Show that the semidirect product H \rtimes Ｚ is isomorphic to a
φ 3
Sylow 3-subgroup of S .
9
5. (ⅰ) Let p be a prime number. Prove that the number of Sylow p-subgroups in
the symmetric group S is exactly (p-2)!.
p
(ⅱ) Apply the 3rd theorem of Sylow to prove the following (Wilson)
congruence for prime numbers p:
(p-1)! ≡ -1 (mod p)
6. (ⅰ) Let p be a prime number dividing |G| for a given finite group G. Let
n (G) denote the number of Sylow p-subgroups in G. Given normal subgroup
p
N \unhld G. Show that if P is a Sylow p-subgroup of G, then PN/N is a
Sylow p-subgroup of G/N.
(ⅱ) Prove n (G/N) ≦ n (G) always holds.
p p
/*
註: 有些符號因為打不出來(應該說打出來也不好看)，只好用latex上的寫法來呈現:
～
(ア) \cong: ＝
(イ) \rtimes: ╳▏
(ウ) \unhld: ＜▏
￣
*/