Let G be a group (written additively), S a nonempty set and M (S, G) the set of all functions from S in G. We define addiction in M (S, G) as follows: For ?, ? ∈ ? (S, G) and for each s ∈ S, (f + g) (s) = f (s) + g (s). Prove that M (S, G) is a group with this operation, and that it is abelian if and only if G

Let G be a group (written additively), S a nonempty set and M (S, G) the set of all functions from S in G. We define addiction in M (S, G) as follows: For ?, ? ∈ ? (S, G) and for each s ∈ S, (f + g) (s) = f (s) + g (s). Prove that M (S, G) is a group with this operation, and that it is abelian if and only if G

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

See similar textbooks

Similar questions

Let Z denote the centre of a group G. Then show that z € Z iff N(a) = G. Also show that if G is finite, then a € Z iff o(N(a)) = o(G).
KS A group homomorphism f:G G'is called Epimorphisum if f is 1-1 False True
Find all group homomorphisms from Z₂ XZ₁ to Z₂ XZ₁ 3 4 2 3 Find all group homomorphisms from 7/3 X 7/4 to 1/₂ X 1/3
(4) Chapter 2, exercise 6.9: Prove that a group and its opposite group Gº (Exercise 2.6) are isomorphic.
Which of the following groups are cyclic? Justify. (a) G = U(10) = {k € Z10 : gcd(k, 10) = 1} = {1,3, 7,9} under multiplication. (b) (R, +).
16. Give a specific example of some group G and elements g, h = G where (gh)" # gh".
19. 20. 21. 22. 23. 24. 25. Consider the groups U(8) and Z4 c) d) e) f) Let H₁, H₂ and H, be abelian groups. Prove or disprove: H, xH₂x H, is an abelian group. List the elements of U(8) x Z4 Determine the identity element of this group. Determine all the elements which are of order two in this group. Determine all the elements of order 4 in this group.? Determine the subgroup generated by the element (7,1) Let K = {xe G|xg=gx Vge G} Show that K is a normal subgroup of G. Define f: R² R² by f(x,y) = (x + 2y, 0) h) i) a) b) Show that f is a homomorphism from into itself Find Ker(f) Let 0: - be defined by 0 la b C by σ(n) =i" a) b) Prove that is a homomorphism Determine Ker (0) Let G= - Verify that o is a homomorphism Find Ker(0) (a,b,c,de R ) Use the Fundamental homomorphism theorem to prove that 45
Theorem 3. Let G be the external direct product of groups G, G, ... G Let then G, x G, x..x G,-xG x...x G, G, (i) If x e H, y e H, for same i #j, then xy= yx.
1. Let G be a group and let H, H, .., H, be the subgroups of G Then G is an internal direct product of H1, H, . H, if and only if the following conditions are satisfied : y.... H; 4G for i = 1, 2, .., n ..... (i) H;n IIH, ={e} jti (ii) G = H, H,.. H,
3.42. Let G be the set of all sequences of integers (a₁, a2, a3, . . .). 1. Show that G is a group under (a₁, a2, ...) + (b₁, b₂, ...) = (a₁ + b₁, a₂ + b₂,...). 2. Let H be the set of all elements (a₁, a2, ...) of G such that only finitely many a are different from 0 (and (0, 0, 0, ...) E H). Show that H is a subgroup of G.
(b) Let K be the set of integers with the operation o defined by xoy=x+y + 1. Prove that K with the operation o is a group.