cture handwriting Q1) Consider the additive group G = (0,1,2,3,4,5,6,7) under modulo-8 addition. Let II (0,2,4,6 } be a sue of G. a) Does H form a subgroup of G? Justify your answer. b) Find all the distinct cosets of H. Q2) Determine whether cach of the following polynomials is irreducible over GF(2). Justify your answer. a) f(x) = 1+ x? + x3 b) f(x) = 1+x2 +x* 6.

cture handwriting Q1) Consider the additive group G = (0,1,2,3,4,5,6,7) under modulo-8 addition. Let II (0,2,4,6 } be a sue of G. a) Does H form a subgroup of G? Justify your answer. b) Find all the distinct cosets of H. Q2) Determine whether cach of the following polynomials is irreducible over GF(2). Justify your answer. a) f(x) = 1+ x? + x3 b) f(x) = 1+x2 +x* 6.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 12E: 12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units...

See similar textbooks

Similar questions

Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.
Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .
Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.
12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.
16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.
15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .
Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.
Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
4. Prove that the special linear group is a normal subgroup of the general linear group .
Describe all subgroups of the group under addition.
9. Find all homomorphic images of the octic group.
let Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n are relatively prime. Exercise16 For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication.