課程名稱︰代數導論二
課程性質︰系定必修
課程教師︰莊正良 教授
開課系所︰數學系
考試時間︰2005/05/04
試題：
1.(15%) For real numbers a≠0 and b, let f_a,b denote the transformation
from reals defined by x → ax＋b.
(i) Show that the set G of transformations f_a,b forms a group under
composition.
(ii) Show that the map θ: f_a,b → a is a group homomorphism from G
onto the multiplication group of nonzero reals.
(iii) Find the kernel of θ.
2.(20%) Let G be a group.
2 2 2
(i) If (ab) ＝a b for all a,b∈G, prove that G is abelian.
3 3 3 5 5 5
(ii) If (ab) ＝a b and (ab) ＝a b for all a,b∈G, prove that G is
abelian.
3.(15%) Let G be a group and a∈G.
(i) Define the centralizer C(a) of a in G.
(ii) Prove that C(a) forms a subgroup of G.
(iii) Assume that G is finite. Prove that the number of conjugates of a
is equal to [G : C(a)].
4.(20%) def.╭ 1 2 3 4 5 6 ╮
(i) Decompose g ＝ ╰ 6 5 2 4 3 1 ╯into a product of disjoint cycles.
(ii) Find the order of g.
(iii) Consider g as an element of S_n, n≧6, by postulating g(i)＝i for
i＞6.
(iv) Find the centralizer og g in S_n.
5.(15%) Let G be a group that has only a finite number of subgroups. Is G
necessarily finite? Prove it or give a counter example.
6.(15%) Let G be a finite group.
(i) If the order of x∈G is an odd number, prove that there exists
y∈G such that x＝y^2.
(ii) If the order of G is an odd number, prove that any x∈G can be
written in the form x＝y^2 for some y∈G.
(iii) Conversely, if any x∈G is of the form x＝y^2 for some y∈G,
prove that the order of G must be odd.
_ def. _
7.(15%) Let Ｚ[√7] ＝ {a＋b√7: a,b∈Ｚ}, where Ｚ denotes the ring of
intergers. _
(i) Find the unit u∈Ｚ[√7] such that u＞1 and such that u is minimal
possible.
(ii) Does the equation x^2－7y^2＝－1 have integer solutions? Give your
reason.
(iii) Does the equation x^2－7y^2＝5 have integer solutions? Give your
reason.
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