MAT 301

MAT 301 - Problem Set 4 Due date Dec 7, 2015 at 4:30, Bahen 6266 1. Let G be a group that contains a normal subgroup H such that |H| = 5 and the factor group G/H is isomorphic to D3. Prove the following: a) G contains a normal subgroup H1 such that |H1| = 15. b) G contains distinct subgroups H2, H3 and H4, each of order 10, none of which is a normal subgroup of G. (Hint: As mentioned in class, the function φ : G → G/H defined by φ(a) = aH, a ∈ G, is onto and is a homomorphism with kernel H. 2. For the groups G and H defined below, prove that H is a normal subgroup of G and the factor group G/H is isomorpic to the group R ∗ of nonzero real numbers (under multiplication). Let G =  A =  a b 0 d  | a, b, d ∈ R, ad 6= 0  under matrix multiplication H =  A ∈ G | A =  a b 0 a  , a 6= 0  . 3. For each group G, prove or disprove that there exists a homomorphism φ : G → Z2⊕Z2 such that φ is onto. Note: Feel free to use the fact that a noncyclic group of order 4 is isomorphic to Z2 ⊕ Z2. a) G = Z. b) G = D10. 4. Let G = D5 ⊕ Z30. Find a subgroup H of order 12 in G. Is H normal in G? Is H cyclic? (Explain.) 5. Let G1 = D6 ⊕ Z3, G2 = S3 ⊕ Z6, G3 = D3 ⊕ D3 and G4 = Z3 ⊕ A4. The group G1 is isomorphic to one of groups G2, G3 and G4. Determine which one by elimination. Explain your answer. (Note: It is not necessary to define an isomorphism.) 6. Let G be an abelian group. a) Let ` be a positive integer. Let H` = { a ∈ G | a ` = e }. Prove that H` is a subgroup of G. For parts b)–d), assume that m ≥ 2 and n ≥ 2 are positive integers and G contains elements of order m and elements of order n. b) Define φ : Hm ⊕ Hn → G by φ(a, b) = ab, for a ∈ Hm and b ∈ Hn. Prove that φ is a homomorphism. Page 1 of 3 c) Prove that φ is one-to-one if and only if gcd(m, n) = 1. d) Prove that if gcd(m, n) = 1, then Hm ⊕ Hn is isomorphic to a subgroup of G. Part e) is not to be handed in. This can be used as a first step in the theorem which classifies finite abelian subgroups in terms of external direct products of cyclic groups Gj of prime power order. Note that the groups Gj are not necessarily cyclic. The second part of the proof (which is discussed in the text) would involve expressing an abelian group of prime power order as (isomorphic to) an external direct sum of cyclic groups of prime power order. e) For this part, assume that G is finite (and abelian). Let r be the number of distinct primes that divide |G|. Prove that G is isomorphic to an external direct product G1 ⊕ G2 ⊕ · · · ⊕ Gr, where Gj is an abelian subgroup of G, and |Gj | is a power of a prime number pj , 1 ≤ j ≤ r, and, if r > 1, pi 6= pj for i 6= j. 7. Let α = (1 4 6 3 7)(2 5 3 8)(4 7 8 1) ∈ S8. a) Determine how many elements in the subgroup hαi are conjugate to α. (Explain your answer.) b) Find β ∈ S8 such that βαβ−1 = α −1 . c) How much can you find out about the disjoint cycle form of the various elements γ ∈ S8, such that γαγ−1 = α −1 ? Can such an element γ belong to A8? (Note: It is possible to compute all possibilities for γ by brute force without too much work. If you don’t want to do that, it is okay to provide a prediction for the form of γ and describe the general ideas used to make your prediction.) 8. Let p and q be odd primes such that p < q. Suppose that G is a nonabelian group of order pq. (Such a group exists whenever p divides q − 1. This fact is not needed in the solution.) a) Prove that Z(G) = {e}. b) Prove that if a ∈ G and a 6= e, then hai = C(a). (Here, C(a) is the centralizer of a in G.) c) Without using Sylow’s First Theorem or Cauchy’s Theorem, use properties of the class equation for G to prove that G contains elements of p and elements of order q. d) Suppose that |G| = 21 (and G is nonabelian). Determine the number of conjugacy classes in each of the sets { a ∈ G | |a| = 3 } and { a ∈ G | |a| = 7 }. (Hint: This can be done using the class equation for G.) Page 2 of 3 More challenging question: (Not to be handed in) Suppose that G is a group having a normal subgroup H such that H is isomorphic to D3. Prove that there exists a subgroup K of G such that G is isomorphic to H ⊕ K. (Hint: Try K = { a ∈ G | aha−1 = h ∀ h ∈ H }.) Page 3 of 3