課程名稱︰代數導論一
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）︰2011/11/24
考試時限（分鐘）：190 mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Write your answer on the answer sheet. We give partial points. In this examina-
tions, G denotes a group, H, K, denote subgroups of G, and e always is the ide-
ntity of G. Also, Α(G) denote the group of all automorphisms of G, and N(H) =
-1
{ x∈G | xHx = H }
(1) (5 points each) "yes" or "no". Either give a brief reason or give a counter
example.
(a) If both H and K are normal subgroups of G with H∩K = (e), then H com-
mutes with K.
(b) The order of an element of the symmetric group S is at most 7.
7
(c) The alternating group A is simple.
3
(d) The alternating group A is simple.
4
(e) The number of conjugates of (1,2,3) in S equals 20.
5
(f) If o(G) is an even natural number, then there exist some x∈G, x≠e
such that the number of conjugates of x in G is odd.
(2) (15 points) Suppose H is cyclic of order 9.
(a) Prove that Α(H) is of order 6 (10 points).
(b) Prove that if K is a finite group of order relatively prime to 6, then
K commutes with H (5 points).
(3) (20 points) Suppose o(G) = pq where p and q are prime numbers with p > q
and let P and Q are respectively p-Sylow and q-Sylow subgroups of G.
(a) Show that P is normal in G (5 points).
(b) Show that if p≠1 (mod q), then Q is normal in G (5 points).
(↑是"≡"同餘的否定，因為符號打不出來)
(c) Show that if o(G) = 35, then G is cyclic (10 points).
(4) (10 points) Show that if H is a p-Sylow subgroup of G, then N(N(H)) = N(H).
(5) (a) Prove that for n≧3, the subgroup of S generated by all 3-cycles
n
in A (10 points).
n
(b) Prove if G = S and H is a normal subgroup of G containing a 3-cycle,
n
then H contains A (5 points).
n
(6) (10 points) If o(G) = p^n where p is a prime number, prove that there exist
subgroups N , i = 0, 1, ..., r, for some r, such that G = N ⊃= N ⊃= ...
i 0 1
(⊃=：包含或等於)
⊃= N = (e) where N is a normal subgroup of N and where N /N is
r i i-1 i-1 i
abelian.
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