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Determine which of the following are groups under the stated operations. For those which are groups, show that the axioms hold and state the identity and inverses. For those which are not, show that one of the axioms fails. i. The pair ({2, 4, 6, 8), o) where xo y = xy mod 10. ii. The set {a+b√√3|a, b € Z} under addition. iii. The set of all vectors in R³ and the operation is vector product.

Determine which of the following are groups under the stated operations. For those which are groups, show that the axioms hold and state the identity and inverses. For those which are not, show that one of the axioms fails. i. The pair ({2, 4, 6, 8), o) where xo y = xy mod 10. ii. The set {a+b√√3|a, b € Z} under addition. iii. The set of all vectors in R³ and the operation is vector product.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Determine which of the following are groups under the stated operations. For those which are groups, show that the axioms hold and state the identity and inverses. For those which are not, show that one of the axioms fails. i. The pair ({2, 4, 6, 8), o) where x oy = ii. The set {a +b√3|a, b = Z} under addition. iii. The set of all vectors in R³ and the operation is vector product. xy mod 10. Show that the following are homomorphisms. Calculate their kernel and image, showing your working. Giving reasons for your answers, which of the homomor- phisms are injective and which are surjective? i. y: 27 → Z7 given by (x) = 3x mod 7. ii. p: G→ (R*, x) where G = = { (89) | a € P ER} plication and p(A) = det (A) for A E G. (Recall that R* = R\{0}.) iii. Z→ S3 given by (n) = (123)" where S3 is the group of permutations on 3 objects. ER*,bER with matrix multi-
Transcribed Image Text:Determine which of the following are groups under the stated operations. For those which are groups, show that the axioms hold and state the identity and inverses. For those which are not, show that one of the axioms fails. i. The pair ({2, 4, 6, 8), o) where x oy = ii. The set {a +b√3|a, b = Z} under addition. iii. The set of all vectors in R³ and the operation is vector product. xy mod 10. Show that the following are homomorphisms. Calculate their kernel and image, showing your working. Giving reasons for your answers, which of the homomor- phisms are injective and which are surjective? i. y: 27 → Z7 given by (x) = 3x mod 7. ii. p: G→ (R*, x) where G = = { (89) | a € P ER} plication and p(A) = det (A) for A E G. (Recall that R* = R\{0}.) iii. Z→ S3 given by (n) = (123)" where S3 is the group of permutations on 3 objects. ER*,bER with matrix multi-
Determine, giving reasons, which of the following groups are cyclic. i. The subgroup pZ of Z where p is a prime and the operation is addition. ii. The direct product K = Z5 × Z with the usual operation on direct products. iii. The direct product K = Z21 X Z49 with the usual operation on direct products. iv. The subgroup T = {z € C* : |z| = 1} of (C*, ×). i. Write the following permutations in the symmetric group S₁0 as products of disjoint cycles: f 9 - 1 2 3 4 5 6 7 8 9 10 2 3 7 6 8 9 1 10 4 5 2 3 4 5 6 7 8 9 10 10 1 5 7 4 2 8 9 1 3 6 Calculate f-¹ and fg, as products of disjoint cycles. ii. State a criterion for members of a finite symmetric group to be conjugate, and determine which of f, g, f¹, fg from part (i) are conjugate. Let G be a group and 4: G → G be a map defined by y(g) = g¯¹. i. Prove that is an bijection. ii. Prove that is an isomorphism if and only if G is Abelian.
Transcribed Image Text:Determine, giving reasons, which of the following groups are cyclic. i. The subgroup pZ of Z where p is a prime and the operation is addition. ii. The direct product K = Z5 × Z with the usual operation on direct products. iii. The direct product K = Z21 X Z49 with the usual operation on direct products. iv. The subgroup T = {z € C* : |z| = 1} of (C*, ×). i. Write the following permutations in the symmetric group S₁0 as products of disjoint cycles: f 9 - 1 2 3 4 5 6 7 8 9 10 2 3 7 6 8 9 1 10 4 5 2 3 4 5 6 7 8 9 10 10 1 5 7 4 2 8 9 1 3 6 Calculate f-¹ and fg, as products of disjoint cycles. ii. State a criterion for members of a finite symmetric group to be conjugate, and determine which of f, g, f¹, fg from part (i) are conjugate. Let G be a group and 4: G → G be a map defined by y(g) = g¯¹. i. Prove that is an bijection. ii. Prove that is an isomorphism if and only if G is Abelian.
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