MODERN ALGEBRA (MATH 210) PROBLEM SET 6

1. If q is prime, show that the automorphism group of the group ℤ/q is isomorphic to ℤ/(q − 1).

2. Suppose p and q are primes such that p|q −1. Show the existence of a nonabelian group of order pq. (Hint: try a semi-direct product.)

3. If p is prime, prove that every group G of order 2p is either cyclic or isomorphic to D_{2p}. Hint: By Cauchy’s theorem, G must contain an element of order p, and ⟨a⟩ is normal in G because it has index 2.

Problem 2 is part of the classification of groups of order pq, where p < q are prime. The rest is given by: (i) If p doesn’t divide q − 1, then all groups of order pq are cyclic. (ii) If p divides q − 1, there is only one nonabelian group of order pq. Problem 3 is a special case of (ii).

4. Suppose G is a finite group of order mn, gcd(m, n) = 1. Suppose K is a normal subgroup of order m. Then K has a complement iff G has a subgroup of order n.

5. (Using semidirect products to see interesting phenomena.) Suppose G is a finite group, and you want to show that g normalizes A ≤ G, i.e. that g^{−1}Ag = A. Then it suffices to show that g^{−1}Ag ⊂ A. Show that this isn’t true for infinite groups (i.e. that it is possible to have g^{−1}Ag ⊊ A) as follows. Let K = ℚ, and Q = ℤ = ⟨x⟩. Let A ⊂ K be given by ℤ ⊂ ℚ. Choose a ϕ : Q → Aut K so that if G = K ⋊_{ϕ} Q, xAx^{−1} ⊊ A.

6. Show that the splitting field of x^{3} − 3x + 1 is degree 3 over ℚ. (Hint: If α is a root, show that 1/(1 − α) is also a root.)

7. Show that if an irreducible cubic in ℚ[x] has two complex roots and one real root, then its splitting field is a degree 6 extension of ℚ.

8. (In preparation for showing the insolvability of the quintic.) Prove that if σ is a 5-cycle and τ is a transposition in S_{5}, then S_{5} is generated by {σ, τ}.

The set is due Tuesday, November 26 at 3:30 pm in Pierre Albin’s mailbox.

Date: Tuesday, November 19, 2002.