[試題] 98上 朱樺 代數導論 期中考
課程名稱︰代數導論
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（年月日）︰98/11/19
考試時限（分鐘）：150min
是否需發放獎勵金：yes
（如未明確表示，則不予發放）
試題 :
(1)(10%)Solve the follow system of equations
┌5x^2 + 4x ≡5 (mod7)
│
└4x ≡6 (mod22)
(2)(10%)Let m >= 2 be an integer and S = { a:gcd(a,m) = 1 , 1 <= a <= m }.
Let n = (a的連乘 其中a屬於S) show that n≡ +-1(mod m).
(3)(10%)(a) Determine the number of elements of order 6 in S7.
(b)Determine the number of odd permutations of order 6 in S7.
(4)(10%)Show that the symmetric group Sn, n>=3, can be generated by the
permutations (1 2) and (1 2....n).
(5)(30%)Let G be a group.
(a) Let |g| = m, |h| = n in G. Suppose that gh = hg and gcd(m,n) = 1.
Show that |gh| = mn.
(b) Let |g| = m, |h| = n in G. Suppose that gh = hg. Show that there
exists a 屬於 G such that |a| = lcm(m,n).
(c) Let |a| = mn in G. Suppose that gcd(m,n) = 1. Show that there exist
g,h 屬於 G such that |g| = m, |h| = n and a = gh.
(6)(30%)(a) Show that for any positive integer n, there is a group G such
that it has exactly n subgroups.
(b) Find a noncyclic group G which has exactly five subgroups.
(c) Let G be a group. Show that G is a finite group if and only if G has
only finitely many subgroups.
(7)(10%) Find all group homomorphisms from the symmetric group S3 to
the Klein 4-group K4
(8)(10%) Let H and K be subgroups of G. Show that Ha∩Ka = (H∩K)a.
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