课程名称︰代数导论一
课程性质︰数学系大二必修
课程教师︰李秋坤教授
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2019/12/20
考试时限（分钟）：50
试题 :
(满分100分)
(以下的属于符号都用ε代替)
1.
(a) (10%) Determine Z(D8), the center of the Dihedral group D8.
(b) (10%) Find all subgroups of D8.
2. (20%) Show that, for each hεH, dimR R[h]≦2.
(Here H is the Quaternions, the 4-dimensional real algebra.)
3.
(a) (10%) Let G be a cyclic group of order n and let dεN. Show that if
d divides n then G contains a subgroup of order d.
(b) (10%) Show that A4 has no subgroup of order 6. (Hint: Suppose that H
is a subgroup in A4 of order 6. Show that there exists a 3-cycle
σ not belongs to H. And then consider H, σH, and (σ^2)H.)
4. (10%) Suppose that m and n be relatively prime positive integers. Determine
whether there exists a non-trivial group homomorphism from the
additive group Zm to the additive group Zn or not, and verify your
answer.
5. Let R and S be rings with identity elements 1R and 1S, respectively.
Suppose that φ: R→S is a nonzero additive map such that φ(xy)=φ(x)φ(y)
for all x,yεR.
(a) (10%) Show that if φ is surjective then φ(1R)=1S.
(b) (10%) Show that if there is an element uεR such that φ(u) is
invertible in S then φ(1R)=1S.
(c) (10%) Show that if S has no zero-divisors then φ(1R)=1S.