16.Let R* be the multiplicative group of non zero real numbers and let S=R-{-1} bethe group under the operation defined byaab=a+b + ab.Define the mapping f: R*-S by f(r) = r - 1. Show that R* = S. 13.Let ne Z* and G=GL(R) the multiplicative group of all invertible nxn matrices overR. Show thatH = {A E G | det A = 1}is a subgroup of G.9

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Answered: 16. Let R* be the multiplicative group…

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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5. (a) Decide whether (Z, -) forms a group where : Z x Z Z is the usual operation of subtraction, i.e. (m, n) m - n. Justify your answer fully. X1 (b) Consider R? = R x R. Elements of R? have the form in which x1, x2 E R. Define the operation : R2 x R2 R? by ) - „). y2 \Y1x2+ y2, Show that (R?, *) does not form a group. Next, find a suitable subset of R? that forms a group with the operation * as given above. Finally, decide with proof whether this group is abelian. Show (R\{-1}, o) forms a group, where for any a, be R\{-1}, (c) a ob = (a+1)(b+1) – 1. Next, find a group element g such that 2og o3 = 5. Is this element unique? Justify your answer.
Let H be any complexes of a group G, then (HK)\power{-1}=K\power{-1}H\power{-1}
8. Let the set G ZXZ and the binary operation on G be given by the rule (a, 6) • (c, d) = (a + r, 6+ d). It is easily verified that the pair (G, ) is a commutative group. a) Show that the mapping f: G- Z defined by fl(a, b)] = a is a homomorphism from (G, ) onto the group (2,+). b) Determine the kernel of this mapping. c) If II = {(a, a) | a E Z}, prove that (H, ) is a subgroup of (G, ), which is isomorphic to (Z,+) under the function f.
(1) Define the product group of (G, o) and (H, x) as the set Gx H = {(g, h): g = G, he H} with composition where (91, h1) (92, h₂) = (910 92, h₁ x h₂). Prove that G x H this is group, and that it is isomorphic to H x G. What is the identity element? What is the inverse of (g, h)?
Kindly answer correctly, Please show the necessary steps 1a
Let (IR,+) be a group of real numbers under addition and (R+,-) be the group of positive real numbers under multiplication. Prove f: R→ R+ by f (x)= ex for all x ER is homomorphism and isomorphism.
(b) Let G be the group generated by x, y, z with the only relation xyxyz-¹. Show that G is a free group.
Let 0:250 Z15 be a group homomorphism with 0(x)=4x. Then, Ker(0)= * O (0, 10, 20, 30, 40} None of the choices (0, 5, 15, 25, 35, 45) O (0,15,30,45} acer
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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.

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16. Let R* be the multiplicative group of non zero real numbers and let S =R-{-1} be the group under the operation defined by aab=a+b+ ab. Define the mapping f: R* S by f(r) = r - 1. Show that R* = S.