课程名称︰代数一
课程性质︰数学系选修 可抵必修代数导论一
课程教师︰林惠雯教授
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰105.10.24
考试时限（分钟）：50分钟
试题 :
1.
Let G act on a non-empty set X and x in X. Show that |Gx| = [G : G_x].
2.
Show that if G is a p-group, then Z_G≠{1} and in particular, if G is a non-abelian group of order p^3, then |Z_G| = p.
3.
Let F be a field. Show that any finite subgroup of (F^x, ‧, 1) is cyclic.
4.
Let G be a finite group and H be a subgroup of G with [G : H] = p being the smallest prime dividing |G|. Show that H is a normal subgroup.
5.
Construst a non-abelian group of order 55.