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.

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Answered: and are groups. Let (2, 9)| E A, y E.…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 5E

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and are groups. Let (2, 9)| E A, y E. and denne the operation on X x Y as (T1, 41) * (x2, Y2) = (x1 0 12, Y1 • Y2) %3D for (21, y1), (*2, Y2) E X × Y. Show that (X x Y, ) is a group.