課程名稱︰代數導論一 課程性質︰數學系必修 課程教師︰林惠雯 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2017 年 11 月 9 日 考試時限（分鐘）：115 分鐘 試題 : * Please write down the key details of your answers. 1. (20%) -1 (a) In S , compute (1 2 3 4 5) (3 6 2) (4 7 1 8) (1 2 3 4 5) . 8 _ × (b) Determine the inverse of 7 in Z . 11 10 (c) In S , compute (2 4 6 8 10 12 14 16 18 20) . 20 _ _ _ __ (d) Find | (Z ×Z ×Z ×Z ) /〈(2, 4, 8, 18)〉|. 4 12 20 24 2 -1 -1 3 3 i j (e) Write the product x yx y x y in the form x y with a rotation x and a reflection y in the dihedral group D . 8 2. (16%) (a) Given an equivalence relation R on S, show that any two equivalence classes are either the same or disjoint. (b) Show that the set of all left cosets of a subgroup H in a group G forms a partition of G. 3. (20%) (a) Derive a formula for computing the number of elements in the orbit of x ∈ X under an action of a group G. (b) How to determine the number of conjugates of an element x in a group G and the number of conjugates of a subgroup H of G? (c) A p-group G acts on a finite set X and X is the subset of 0 X which is fixed by the whole G. Show that |X| ≡ |X | (mod p). 0 4. (16%) (a) Let |G| = 49. Show that G has at least one subgroup of order 7, and that if it contains only one subgroup of order 7, then it is a cyclic group. 2 (b) Show that if |G| = p with p a prime number, then G is abelian. 5. (18%) (a) State and show the Burnside's formula. (b) Choose 8 pearls from pearls of 3 different colors and chain them together to make a necklace. How many different necklaces can one have? 6. (20%) 3X3 (a) Decompose the set C of 3X3 complex matrices into orbits for the following operations of GL (C): 3 (1) left multiplication, (2) conjugation. (b) Find the order of the orbit of the matrix diag(1, 2, 3) under conjugation in GL (F ). 3 7 --

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