Homework Answers
3 Let p and q be prime numbers and let G be a non-cyclic group of...
3 Let p and q be prime numbers and let G be a non-cyclic group of order pq. Let H be a subgroup of G.Show that either H is cyclic or H-G. 12 - Let I and J, be ideals in R. In, the homomorphismJ f: (!+J a → a+J use the First Isomorphism Theorem to prove that I+J
Let n > 1. Suppose that Zn is cyclic. Prove that n must be prime.
Let n > 1. Suppose that Zn is cyclic. Prove that n must be prime.
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if...
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in
Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q
contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q)
is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP +
βQ|α, β ∈ Z[x]}.
(iii) For which primes p and which integers n ≥ 1 is the
polynomial xn − p...
Let G be a finite group such that p is a prime and p divides |G|....
Let G be a finite group such that p is a prime and p divides
|G|. Let P be a p-Sylow subgroup of G such that P is cyclic and ?
. Let H be a subgroup of P . Prove
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Abstract Algebra 1 a) Prove that if G is a cyclic group of prime order than...
Abstract Algebra 1 a) Prove that if G is a cyclic group of prime order than G has exactly two subgroups. What are they? 1 b) Let G be a group and H a subgroup of G. Let x ∈ G. Proof that if for a, b ∈ H and ax = b then x ∈ H. (If you use any group axioms, show them)
8. Let p be a prime number. Define -c0t}cQ ZAp) Prove that Zp) is a subring...
8. Let p be a prime number. Define -c0t}cQ ZAp) Prove that Zp) is a subring of Q Prove that Z is a subring of Z Show that the field of fractions of Zp) is isomorphic to Q
8. (a) Prove that if p and q are prime numbers then p2 + pq is...
8. (a) Prove that if p and q are prime numbers then p2 + pq is not a perfect square. (b) Prove that, for every integer a and every prime p, if p | a then ged(a,pb) = god(a,b). Is the converse of this statement true? Explain why or why not. (c) Prove that, for every non-zero integer n, the sum of all (positive or negative) divisors of n is equal to zero. 9. Let a and b be integers...
12. Let p be a prime. Prove that there is a unique conjugacy class of p-cycles...
12. Let p be a prime. Prove that there is a unique conjugacy class of p-cycles in Ap+2.
Let p and n be integers. Prove that, if p is prime, then gcd(p, n) =...
Let p and n be integers. Prove that, if p is prime, then gcd(p, n) = p or gcd(p, n) = 1. . . (i.) Using proof by contrapositive (ii.) Using proof by contradiction
Let p be a prime with p ≥ 13. Prove that among the integers 2,11 and...
Let p be a prime with p ≥ 13. Prove that among the integers 2,11 and 22, either all three are quadratic residues modulo p or exactly one is a quadratic residue modulo p.
3 Let p and q be prime numbers and let G be a non-cyclic group of order pq. Let H be a subgroup of G.Show that either H is cyclic or H-G. 12 - Let I and J, be ideals in R. In, the homomorphismJ f: (!+J a → a+J use the First Isomorphism Theorem to prove that I+J
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in
Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q
contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q)
is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP +
βQ|α, β ∈ Z[x]}.
(iii) For which primes p and which integers n ≥ 1 is the
polynomial xn − p...
Let G be a finite group such that p is a prime and p divides
|G|. Let P be a p-Sylow subgroup of G such that P is cyclic and ?
. Let H be a subgroup of P . Prove
We were unable to transcribe this imageWe were unable to transcribe this image
8. Let p be a prime number. Define -c0t}cQ ZAp) Prove that Zp) is a subring of Q Prove that Z is a subring of Z Show that the field of fractions of Zp) is isomorphic to Q
8. (a) Prove that if p and q are prime numbers then p2 + pq is not a perfect square. (b) Prove that, for every integer a and every prime p, if p | a then ged(a,pb) = god(a,b). Is the converse of this statement true? Explain why or why not. (c) Prove that, for every non-zero integer n, the sum of all (positive or negative) divisors of n is equal to zero. 9. Let a and b be integers...
12. Let p be a prime. Prove that there is a unique conjugacy class of p-cycles in Ap+2.
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