課程名稱︰代數導論一
課程性質︰系必修
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）︰2010年10月28日
考試時限（分鐘）：三小時
是否需發放獎勵金：是
試題 :
In answering the following problems, you may use freely any Theorem already
proved (or Lemmas, Propositions) from the textbook(Section 2.2-2.6), my class-
room lectures, or previous course on Linear Algebra. You don't need to give
proofs of the theorems you are using, but you MUST write down complete statem-
ent of the theorem which you use as a second part of your answer to that prob-
lem.
NOTATIONS：
Ⅱm ,the cyclic subgroup of order m>=1, known also as the additive group of
the integers modulo m.
D2n ,the dihedral group of order 2n, known also as the group symmetries of the
regular n-gon, n>=3.
Sn , the symmetric group of n>=1 letters.
An , the subgroup of Sn consists of even permutations.
V = {1,(12)(34),(13)(24),(14)(23)} <= S4.
Q, the quaternion group generated by A = [ 0 1 ] and B = [ 0 i ]
[ -1 0 ] [ i 0 ]
under the matrix multipication.
GL(2,R), the group of all real 2 ×2 matrices with non-zero determinant.
SL(2,R), the subgroup of GL(2,R) consisting of matrices having determinant 1.
I, the identity square matrix.
(1)(a) Prove that Ⅱ3 ×Ⅱ9 is not isomorphic to Ⅱ3 ×Ⅱ3 ×Ⅱ3.
(b) Suppose gcd(m,n) = 1, prove that Ⅱm ×Ⅱmn ~ Ⅱn ×Ⅱm ×Ⅱm.
=
(2)(a)Write down all subgroups of D10. Which ones are normal subgroups? Which
ones are non-normal subgroups?
(b)Let n>=2 be integer. Prove that there exist two elements in the group D2n,
both of order 2, such that any subgroup of D2n containing these two ele-
ments must be the whole group D2n (in order words D2n can be generated
by these two elements).
(c)Prve that D12 ~ S3 ×Ⅱ2.
=
(3)(a)Show that S4 has three distinct subgroups of order 8, and they are all
isomorphic_to D8.
(b)Show that V <| S4, and prove that S4/V ~ S3.
=
(4)(a)Let H,K <= G be subgroups of G. Prove that HK is a subgroup if and only
if HK = KH holds.
(b)The following statement is NOT true: let K, H1 ,H2 be subgroups of group
G, with K <| G satisfyinh H1K = H2K , then H1 = H2. Give an example of a
group G, together with subgroups K,H1,H2 satisfying the hypothesis of
this statement, but H1 ≠ H2.
(5)(a)Show that every element x ≠ ±I. Then prove that every subgroup of
Ⅱ3 ×Q is normal.
(b)Suppose G is finite abelian group with |G| odd. Show that for every
divior d of 8|G|, there exists subgroup in G ×Q having order d.
(6)(a)Given x,y in group G, the element xy x^(-1) y^(-1) is called the commut-
ator of x and y. The set of all commutators of pairs of elements form G
generate a subgroup, called the commutator subgroup of G. Show that the
commutator subgroup of GL(2,R) is contained in SL(2,R).
(b)Prove that the commutator subgroup of Sn is contained in An. Show also
that the commutator subgroup of S3 is A3.
以上試題結束。
附註：
(3)(a)只需要找到三個，不必證明只有三個。
另外提供得分分布：
A 12 , A- 6
B+ 1 , B 4 ,B- 3
C+ 2 , C 4 ,C- 1
F 70. （如果你作不太出來並不用太難過，但還是請好好加油吧!)
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