MATH 241 Final, Fall 2005

1 Time: 2 Hours MATHEMATICS 241 Name ------------------ January 25, 2006 FINAL EXAMINATION ID ----------------- Prof. H. Abu-Khuzam. First Semester 2005-2006 1. Suppose that a group G is the internal direct product of its normal subgroups H and K. Prove that G/H ≈ K. [ 9 points] 2. Let G be a finite group of order n. Suppose that ϕ:G → H is a homomorphism from G onto a group H of order m. Prove that m divides n. 2 [ 7 points] 3. Show that the alternating group An ( n >1 ) of all even permutations in Sn has order n!/2 by defining a one-to-one and onto mapping from An onto the set Bn of odd permutations in Sn. [ 7 points] 4. Let G be a group of order pq, where p, q are primes with p<q. Prove that G has a normal subgroup of order q [ 7 points] 3 5. Let K be a cyclic normal subgroup of a group G . Prove that every subgroup of K is normal in G. [ 7 points] 6. Let G be an abelian group. Let a, b ∈ G such that o(a)= p, o(b)= q , where p and q are distinct primes. Show that o(ab) = pq. [ 7 points] 4 7. Prove that each homomorphism from a field to a ring is either one-to-one or maps everything onto {0}. [ 7 points] 5 8. Let R be a finite commutative ring with identity. Prove that every prime ideal in R is a maximal ideal. [ 7 points] 9. (a) Let R be ring with identity element 1R. Let ϕ : R → S be a ring homomorphism from R onto S. Prove that ϕ 1( R ) is the identity of S. [ 6 points] 6 (b) Give an example of a ring homomorphism S R ′ ′ → ϕ : of rings with identity such that ϕ1( R′ ) S ′ 1 [ 6 points] 10. Let R be a commutative ring with identity 1. Let M R be an ideal of R such that R/M is a field. Show that for each M R r r ∉ ∈ , , there exists s ∈ R such that M rs −1∈ [ 8 points] 7 11. Answer TRUE or FALSE only: [ 2 points each, -1/2 point penalty for each wrong answer]. (a) ----- Z5 is isomorphic to a subgroup of S5. (b) ----- Let ϕ:G → H be a group homomorphism . If G is abelian, then H is abelian. (c) ----- If G is a finite cyclic group, then G has a prime order. (d) ----- If H Z ϕ : 7 → is a nontrivial group homomorphism, then ϕ is one-to-one. (e) ----- Let S=       ∈       Z a b c d d c b a : , , , 2 2 2 2 . Then the ideal S is a prime ideal of the ring of 2×2 matrices over Z. (f) ----- The equation 0 2 3 2 = x − x + has exactly 2 solutions in the ring Z6. (g) ----- The subset S ={(a,a)∈ Z × Z | a∈Z} of Z×Z is an ideal of Z×Z. (h) ----- The direct product of two integral domains is an integral domain (i) ----- No group of order 20 is simple (j) ----- The permutation (1,4,5,6)(2,1,5) is a cycle in S6. (k) ----- If R is a ring of order 7 and I ≠R is a maximal ideal in R, then I ={0}. [ 22 points]