Higher Algebra I — Fall 2005

Gerald Hoehn

Problem sheet 3 September 8, 2005

Problem 1: Let G_{1}, …, G_{n} be finite simple groups. Show that G_{1}, …, G_{n} are the composition factors of the direct product G_{1} × ⋯ × G_{n}.

Problem 2: Show that if the automorphism group of the finite cyclic group of order n is itself cyclic that n must be contained in the set S = {2, 4, p^{k}, 2 p^{k}} with p an odd prime and k a positive integer. (Hint: Show Z/nZ ≅ Z/p_{1}^{a_{1}}Z× ⋯ × Z/p_{l}^{a_{l}}Z where p_{1}^{a_{1}} ⋯ p_{l}^{a_{l}} is the prime factorization of n and use this to investigate the φ(n) = φ(p^{a_{1}}) ⋯ φ(p^{a_{l}}) = (p_{1}^{a_{1}−1}(p_{1} − 1)) ⋯ (p_{l}^{a_{1}−1}(p_{l} − 1)) generators of Z/nZ. Identify the automorphism group of Z/nZ with the multiplicative group (Z/nZ)^{×} of residue classes modulo n of the φ(n) nonnegative integers m < n which are relatively prime to n.)

Problem 3: If K is a normal subgroup of the group G and K is cyclic, prove that the commutator G′ is a subgroup of the centralizer C(K) of K. (Hint: Use from Problem 2 that the automorphism group of a cyclic group is abelian.)

Problem 4: Let G be a finite solvable group. Prove that there exists a chain G = N_{0} > N_{1} > ⋯ > N_{n} = {e} of subgroups of G such that N_{i} is normal in G and N_{i}/N_{i+1} is abelian for i = 0, 1, …, n − 1. (Hint: Prove that a minimal nontrivial normal subgroup M of G is necessarily abelian and use induction. To see that M is abelian, let N be a normal subgroup of M of prime index and show that [x, y] ∈ N for all x, y ∈ M. Apply the same argument to gNg^{−1}, g ∈ G, to show that [x, y] lies in the intersection of all G-conjugates of N, and use minimality of M to conclude that [x, y] = e.)

Problem 5^{∗}: Show that the automorphism group of the finite cyclic group of order n is itself cyclic if n is contained in the set S = {2, 4, p^{k}, 2 p^{k}} with p an odd prime and k a positive integer.