課程名稱︰代數導論二
課程性質︰必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/5/15
考試時限（分鐘）：120
試題 :
Part I (60 points) True or false. Either prove the assertion or disprove
it by a counter-example (10 point each):
(1) A group of order 515 is simple.
(2) Every group of order 49 is commutative.
(3) The Sylow 2-group of Z/12z * Z/20Z is Z/4Z * Z/4Z.
(4) All Sylow 7-groups of the symmetric group S_7 are cyclic.
(5) Let V be the abelian group generated by x, y, z, u, with
relations: 2x + 1z + 6u = 0, 5y + z + 5u = 0.
Then V is an infinite group.
(6) The polynomial x^5 + 5x + 15 is irreducible in Q( 5^(1/3) ).
Part II (75 points). For each of the following problems, give accordingly
a short proof or an example (15 points each):
(1) Find two non-isomorphic non-commutative groups of order 8.
(2) If the class equation of G is 1+8+9, then G contains a normal
Sylow 3-subgroup. 3 1 2
(3) Identify the abelian group presented by the matrix ( 1 1 1 ).
2 3 6
(4) Let R be an integral domain that contains a field F as subring
and that is finite dimensional when viewed as vector space over
F. Then R is also a field.
(5) Two abelian groups of order 2016 having the same amount of
elements of order 6 must be isomorphic.