课程名称︰代数一
课程性质︰数学系大二必修
课程教师︰于靖
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2015/11/16
考试时限（分钟）：180
试题 :
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
/*
注: 有些符号因为打不出来(应该说打出来也不好看)，只好用latex上的写法来呈现:
～
(ア) \cong: ＝
(イ) \rtimes: ╳▏
(ウ) \unhld: ＜▏
￣
*/