課程名稱︰代數導論一
課程性質︰數學系必修
課程教師︰朱樺
開課學院：理學院
開課系所︰數學系
考試日期︰2010年01月14日
考試時限：150分鐘
是否需發放獎勵金：Yes
試題 :
(1) (a,10%; b-e,6%; f,11%) Let Q_10 = <x,y : x^10 = y^4 = 1, x^5 = y^2,
yxy = x^(-1)> be a group of order 20.
(a) Partition G into conjugacy classes.
(b) Find all normal subfroups of G.
(c) Find the center Z(G) and the commutatr subgroup G' of G.
(d) Find all Homomorphis images of G.
(e) Find a Sylow 2-subgroup of G.
(f) Find the automorphism group Aut(G).
(2) (15%) Let H,K,N≦G and H 包含於 N.
(a) Show that (HK)∩N = H(K∩N).
(b) If H∩K = N∩K and HK = NK, show that H = N.
(3) (10%) Let N be a normal subgroup of G. If N and G/N are finitely generated
groups, show that Gis also finitely generated.
(4) (10%) Let K⊿G where K is cyclic. Show that every subgroup of K is normal
in G.
(5) (10%) if K⊿G has index m and gcd(m,n)=1 , show that K contains every
element of G of order n.
(6) (10%) Let G be a finite p-group. Show that the number of subgroups of
order p^k is congruent modulo p to the number of normal subgroups of order
p^k.
(7) (10%) If G is a group of order 351, ahow that G is not simple.
(8) (10%) Classify all groups G with │G│ = 21.
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