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 . -- 移居二次元(|R^2)的注意事項： 3. 如果你在從事random walk，往上下左右的 1. connectedness不保證pathwise connec- 的機率都是1/4，則你能回家的機率是1。 tedness。可能你跟你的幼馴染住很近， 4. 下面這個PDE是二次元上的波方程式 卻永遠沒辦法到她家。 http://i.imgur.com/2H9HllP.png 2. ODE的C^1 autonomous system不會出現 它的解不滿足Huygens' principle，因此 chaos，在預測事情上比較方便。 講話時會聽到自己的回音，很不方便。 --

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