Higher Algebra I — Fall 2005

Gerald Hoehn

Problem sheet 6 September 29, 2005

Problem 1 (Abelian Groups): (2 points)

(a) Determine the number of non-isomorphic abelian groups of order 5715360000.

(b) Determine the number of subgroups of order 1000 of Z/1000Z×Z/1000Z.

Problem 2 (Presentations): (6 points)

Prove the following group presentations:

(a) ⟨a, b | a^{2}b^{−2}, a^{−1}bab⟩ = Q_{8}, the quaternion group of order 8 defined on sheet 1.

(b) ⟨τ_{1}, …, τ_{n−1} | τ_{i}^{2} for 1 ≤ i ≤ n − 1; (τ_{i}τ_{i+1})^{3} for 1 ≤ i ≤ n − 2; (τ_{i}τ_{j})^{2} for 1 ≤ i, j ≤ n − 1 with |i − j| ≥ 2 ⟩ = S_{n}, the symmetric group of degree n. (Hint: τ_{i} is the transposition (1 i+1). Use induction on n.)

(c) ⟨a, b, c | bab^{−1}a^{−2}, cbc^{−1}b^{−2}, aca^{−1}c^{−2}⟩ = {e}, the trivial group with one element.

Problem 3 (Coproducts for Abelian Groups): (4 points)

Prove the existence of coproducts for abelian groups, i.e., prove Theorem 39 with each occurrence of the word “group” replaced by “abelian group”.

Problem 4 (Pushouts): (4 points)

Let F, H and G be groups and f : G ⟶ F, h : G ⟶ H be homomorphisms. Prove the existence of a group F ∐_{G} H (called the pushout of f and h) and homomorphisms q_{1} : F ⟶ F ∐_{G} H, q_{2} : H ⟶ F ∐_{G} H with q_{1} ∘ f = q_{2} ∘ g such that the following universal property is satisfied: For any group J and homomorphisms j_{1} : F ⟶ J, j_{2} : H ⟶ J with j_{1} ∘ f = j_{2} ∘ g there exists a homomorphism φ : F ∐_{G} H ⟶ J such that j_{1} = q_{1} ∘ φ and j_{2} = q_{2} ∘ φ.

Problem 5^{∗}: Show that the group ⟨S, T | S^{2}, (ST)^{3}⟩ is isomorphic to the group PSL_{2}(Z) = SL_{2}(Z)/\{±({1 0 \atop 0 1})\}, where SL_{2}(Z) is the group of 2 × 2matrices with integer entries and determinant 1. (Hint: Consider the action of SL_{2}(Z) by Möbius transformations on the upper halfplane H = {z ∈ C | ℑ(z) > 0\} of complex numbers given by (({a b \atop b c}), z) ↦ \frac{az+b}{cz+d}; cf. the book by J.-P. Serre: A course in Arithmetic.)