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

KS A group homomorphism f:G G'is called Epimorphisum if f is 1-1 False True
(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
4*. Let f G H be a group homomorphism. Prove: (a) If S G then f(S) 4 f(G) (b) Show by example that S aG need not imply f(S) (c) If T H then f1(T) G. H
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.
Define * on Q* by a * b = - ab Show that (Q*,*) is a group. 1.1 1.2 Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba. 1.3 Let G be a group and Z(G) = {a € G:ax = xa for all x e G}. Show that Z(G)is a normal subgroup of G. Let G be a group and let p: G → G be the map p(x) = x-1. (a) Prove that o is bijective. (b) Prove that p is an automorphism if G is abelian. Let a, ßE S, (Symmetric group), where a = (1,2)(4,5) and ß = (1,6,5,3,2). Verify that (aß)-1 = B-'a-1 . 1.4 1.5
Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x Y of X and Y, but can we also define a binary operation on X x Y such that X x Y is a group? Let's consider two elements (x1, 41) and (x2, Y2) in X x Y. Let denote a function on X x Y.We need to define so that (X x Y,•) is a group. The most natural way to proceed is to define • component-wise: (T1, Yı) • (x2, Y2) = (x1 º Y1, ¤2 * Y2) %3D Now we should show that (X × Y,•) is in fact a group.