Homework 2 of Algebra I
9/18/2023.
Due on 9/25/2023 in class
請同學使用 A4 紙張，以黑色或藍色筆書寫作業，然後用紅筆對作答的每一題自評分數，加
總後於第一頁上方寫下自評總分，以供助教檢查後登記成績。
1. (10 points) In ⟨Z, +⟩, if a ̸= 0, then ⟨a⟩ ∼
= Z; and if |a| | |a′ | and |a| < |a′ | then ⟨a′ ⟩ ⫋ ⟨a⟩.
2. (10 points) Show that U(8) is isomorphic to U(12).
3. (30th Putnam Competition (1969) Problem B2)
(a) (10 points) Let H and K be two subgroups of a group G such that any one is not
contained in the other. Show G ̸= H ∪ K.
(b) (10 points) Find a group G with three proper subgroups H1 , H2 , H3 of G such that
G = H1 ∪ H2 ∪ H3 .
4. Let D4 = ⟨σ , τ ⟩ denote the dihedral group, where σ 4 = e = τ 2 and τσ = σ −1 τ .
(a) (10 points) How many subgroups of Dn . List all of them.
(b) (5 points) Is there a non-abelian proper subgroup of D4 ?
(c) (5 points) Is there a non-cyclic proper subgroup of D4 ?
5. (a) (10 points) If ⟨G, +⟩ is an abelian group and G = ⟨a, b⟩ is a group with generators a
and b, show that
G = {na + mb | n, m ∈ Z}.
(b) (10 points) Show that ⟨Q, +⟩ is not generated by a finite set.
6. (20 points) Let G be a group with more than p − 1 elements of order a prime p. Show
that G is not cyclic.
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