Math 551, Assignment 5

Due Tuesday, October 24, in class

1. A group K is said to be complete if and only if the natural mapping K → Aut(K) (g ↦ Int(g) ∀g ∈ K) is an isomorphism. Show that if G is a group, K ◁ G and K is complete, then G = K × H for some subgroup H ≤ G.

2. Show that SL_{2}(2) and Σ_{4} are complete, but P SL_{2}(R) and A_{4} are not complete.

3. Show that G = Z_{16} ⊕ Z_{2} has a subgroup H such that H ≅ Z_{8} and G/H ≅ Z_{4}.

4. In this problem, G is an abelian group. The subgroup H ≤ G is said to be pure in G if and only whenever g ∈ G, n ∈ Z and ng ∈ H, there is h ∈ H such that ng = nh. The subgroup H is a direct summand of G if and only if G = H ⊕ K for some subgroup K ≤ G. Show that

a) Any direct summand of G is pure in G.

b) If G has exponent p^{n}, p prime, and and x ∈ G is an element of order p^{n}, then ⟨x⟩ is pure in G.

c) If H is pure in G, then any element of G/H of order m has a preimage in G which is also of order m.

d) If G is finite and H is a pure subgroup of G, then H is a direct summand of G.

5. For any group G, define the derived series

G = G^{(0)} ≥ G^{(1)} ≥ G^{(2)} ≥ ⋯ ≥ G^{(n)} ≥ ⋯

of G inductively by G^{(0)} = G and G^{(i+1)} = [G^{(i)}, G^{(i)}] for each i ≥ 0. Also set G^{(∞)} = ∩_{i=0}^{∞}G^{(i)}.

a) Show that G is solvable if and only if G^{(n)} = 1 for some n. (The least such n is called the derived length of G.)

b) Show that for a finite group G, the projection G → G/G^{(∞)} is a universal homomorphism to a solvable group (in a sense which you are to formulate).

c) Construct a nonsolvable group G for which G^{(∞)} = 1. (Hint. Try an infinite-dimensional version of the matrix group B.)

6. List all the characteristic subgroups (not just their isomorphism types!) of Z_{p^{2}} ⊕ Z_{p}.

7. Let G = G_{1} ×G_{2}, with ρ_{i} : G → G_{i} the natural projection, i = 1, 2. Given any subgroup H ≤ G, show that H ∩ G_{i} ◁ ρ_{i}(H) for each i, and define, using H in a natural way, an isomorphism

ϕ_{H} : ρ_{1}(H)/(H ∩ G_{1}) ≅ ρ_{2}(H)/(H ∩ G_{2}).

Set up a one-to-one correspondence between the set of all subgroups of G and the set of all (ordered) quintuples (A_{1}, B_{1}, A_{2}, B_{2}, ϕ) such that B_{i} ◁ A_{i} ≤ G_{i} for each i, and ϕ : A_{1}/B_{1} → A_{2}/B_{2} is an isomorphism. Use your correspondence to count the number of non-abelian subgroups of Σ_{3} × Σ_{3}.