课程名称︰代数导论二
课程性质︰必修
课程教师︰陈其诚
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/3/24
考试时限（分钟）：120
试题 :
Part I (30 points). True or False. Either prove the assertion or disprove
it by a counter-example (6 point each):
(1) A group of order 21 is not simple.
(2) Every commutative group of order 8 is cyclic.
(3) Every order 21 subgroup of the S_7 is commutative.
(4) Every group of order 423 is simple.
(5) The quotient G/Z(G), where Z(G) denotes the center of G, can
not be cyclic, unless G is commutative.
(6) The class equation of a group of order 2016 can be
1 + 1 + 2 + 3 + 4 + 31 + x.
Part II (40 points). For each of the following problems, give accordingly
a short proof or an example (8 point each):
(1) Find two non-isomorphic non-commutative groups of order 8.
(2) The class equation of a group of order 8 can not be
1 + 1 + 1 + 1 + 4.
(3) If G and G' are both non-commutative groups of order 10, then
G is isomorphic to G'.
(4) If C(x) denotes the conjugacy class of some x in a group of
order 55 such that |C(x)| = 11, then x is of order 5.
(5) Let T be a Sylow 3-subgroup in a group G of order 105. If
T is a normal subgroup of G, then T ⊂ Z(G).
(6) If the class equation of G is 1 + 4 + 5 + 5 + 5, then G contains
a normal Sylow 5-subgroup.
Part III (30 points). Give a complete proof (10 points each):
(1) Suppose a finite group G operates non-trivially on a finite set
S of order r. If |G| > r!, then G is not simple.
(2) A group of order p^2, p a prime number, is commutative.
(3) Let p be a prime number and let G be a p-group. Let H be a
proper subgroup of G. Prove that the normaliser of H is strictly
larger than H.
(4) Let G_1 ⊂ G be groups whose orders are divisible by p, and let
H_1 be a Sylow p-subgroup of G_1. Proves that there exists a
Sylow p-subgroup H of G such that H_1 = H ∩ G_1.