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）︰2015/12/14
考試時限（分鐘）：180
試題 :
In answering the following problems, please give complete arguments as much as
possible. You may ask for any definition. 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. Give an example of finite group G together with a normal subgroup H, such
that H is not a characteristic subgroup of G. In other words there exists
automorphism σ of G with σ(H) ≠ H. Verify your answer. (Hint: try group G
with |G| = 8.)
2. (a) Let |F be the field of two elements. Let G := GL (|F ) be the group
2 n 2
consisting of invertible n ×n-matrices with entries from |F . Compute
2
the order |G|. Show that the subgroup U consisting of strictly upper
n
triangular matrices is a Sylow 2-subgroup of G. (U := {(a ) ∈ G| a
n ij ii
= 1 for 1 ≦ i ≦ n, a = 0 if 1 ≦ j < i ≦ n}).
ij
(b) Show that any group of order 16 is isomorphic to a subgroup of U .
16
(Hint: may use Cayley's theorem which says that a group of order 16 is
isomorphic to a subgroup of the symmetric group S of degree 16.)
16
3. (a) Show that the symmetric group S has 6 Sylow 5-subgroups. Then prove that
5
the group S has a subgroup H which acts transitively on the set {1, ...,
6
6} and is isomorphic to S .
5
(b) Let S act by right multiplication on the six rights cosets Hg in S .
6 6
This gives a permutation representation φ: S → Sym , with A := {Hg}.
6 A
Show that φ then gives an automorphism of S which maps the transitive
6
subgroup H to the group of permutations fixing the element H inside A. It
follows that φ corresponds to an outer automorphism of S .
6
4. (a) Show that the symmetric group S of degree n is generated by
n
transpositions r := (i, i+1), for i = 1, ..., n-1.
i
(b) Let H \cong S be the subgroup of S permuting the set {1, ..., n-1}.
n-1 n
Show that the group S can be decomposed into (disjoint union of) double
n
cosets H ∪ Hr H.
n-1
5. Let F be a field. Take a lexicographic order on monomials in x, y, and z with
2 3
y > z . x. Show that G := {y - x , z - x } is a Gröbner basis for the ideal
2 3 3
I := (y - x , z - x ) ⊂ F[x, y, z]. Also check whether the polynomial x +
3
y + zx + 1 is inside this ideal I.
6. Let F be a field, and f , ..., f ∈ F[x , x , ..., x ] be nonzero
1 m 1 2 n
polynomials in the variables x , ..., x . Given nonzero polynomial f ∈ F[x ,
1 n 1
n
x , ..., x ] satisfying f(a , ..., a ) = 0 whenever (a , ..., a ) ∈ F and
2 n 1 n 1 n
f (a , ..., a ) = 0 for i ≦ i ≦ m. Let y be another variable, and set I ⊂
i 1 n
F[x , ..., x , y] to be the ideal (f , ..., f , 1 - yf). Prove that there is
1 n 1 m
no solution to the system of m + 1 equations f = f = ... = f = 0, 1 - yf =
1 2 m
n+1
0 inside F .