3. Define an operation on G = R\{0} x R as follows:(a, b) (c,d) = (ac, bc + d) for all (a, b), (c,d) € G.Show that (G, *) is a group which has infinitely many element oforder 2.4. Let (G, *) be a group and a, b E G. Show that(a). o(a) = o(a-¹),(b). o(a) = o(b¹ *a*b),(c). o(a*b) = o(b* a).ed Ati

3. Define an operation * on G=ℝ\{0} ×ℝ as follows: (a,b)*(c,d)=(ac, bc+d) for all (a,b), (c,d) ∈G…

Answered: 3. Define an operation on G = R\{0} x R…

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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Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D %3D where a = (1 2 3 4) and b = (2 4). Compute (ab)(ab) Compute (ab)(a²b) Compute (ab)(a³b)
34. Consider the action of S, the symmetric group of order 4, on Z[x,, x2, X3, X4] given by Oe S,. Let Hc S, denote the cyclic subgroup generated by (1423). Then, the cardinality of the orbit O,(x,x3+xx) of H on the polynomial x,x, + x,X4 is (a) 1 (c) 3 (b) 2 (d) 4
2.52. Let G be a group and Z the center of G. If T is any auto- morphism of G, prove that T(Z) CZ.
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.
Write out the Cayley table for the group {1, -1, i, -i} under multiplication where i = sqrt(-1). (In general, we apply the row element on the left of the column element.)
1. Assume (X, o) and (Y,) are groups. Let X x Y = {(r, y)|x E X,y E Y} and define the operation * on X x Y as (11, Yı) * (#2, Y2) = (x1 0 F2, Y1 • Y2) for (r1, y1), (r2, Y2) E X x Y. Show that (X x Y, *) is a group.
6. Additional Classifications: Classify the factor group according to the fundamental theorem of finitely generated abelian groups: a. (Z₁ × Z₁2)/((3,6)) b. (Z x Z)/((3,6))
5. G (a + b.2: a eZ and b e Z) is a group under addition in the reals. Define o : G -→G for all a + b.2 e G, as 0(a+ b.2) = a- b./2. Prove that o is an automorphism.
If ring (R1, +, *) is isomorphic to ring (R2, +, *) with f being the isomorphism,then prove the following statements:a. f(e1) = e2, where e1 is the identity element for the group (R1, +) and e2 isthe identity element of the group (R2, +).b. f(-a) = -f(a), for each element ‘a’ in R1, where –a represent the additiveinverse of element ‘a’ in R1 and –f(a) represents the additive inverse of theelement f(a) in R2.c. If (R1, +, *) is a ring with (multiplicative) identity I1 and (R2, +, *) is a ringwith (multiplicative) identity I2, then f(I1) = I2.
Consider the following map defined on the symmetric group S3: Þ: S3 → GL₂(R) given by: Þ(e) = Þ((123)) = Þ((132)) = (19) ((12)) = ((13)) =((23))= -29 20 -42 29 (a) Show that map & defines a representation of S3.
9. a) find two integers x,y such that 30x+101y=1. (hint: gcd(30,101)=1). b) find the inverse of 30 in the group U(101).
! Let U(12) be the set of all positive integers less than 12 and relatively prime to 12. Find the order of the group (U(12), ®12).

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3. Define an operation on G = R\{0} x R as follows: (a, b) (c,d) = (ac, bc + d) for all (a, b), (c,d) = G. ★ Show that (G, *) is a group which has infinitely many element of order 2.