answer number 2 1. Assume (X, o) and (Y, •) are groups. Let X x Y = {(r, y)|1 € X, y € Y} and define the operationon X x Y as(F1, 4ı) * (x2, Y2) = (#1 0 r2, Y1 • Y2)%3Dfor (21, y1), (x2, Y2) e X × Y. Show that (X x Y, *) is a group.2. Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group.%3D(a) Give its Cayley Table.(b) What is [(a, 1) * (b, 1)]1? what is its order in V x Z2? Justify your answers.

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Answered: 2. Deduce from 1 that V x Z2 is a group…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 31E: Exercises 31. Let be a group with its center: . Prove that if is the only element of order in ,...

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2. Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group. (a) Give its Cayley Table. (b) What is [(a, 1) * (b, 1)]¬1? What is its order in V x Z2? Justify your answers.