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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