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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.

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.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.1: Definition Of A Group
Problem 45E: 45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )
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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.
Transcribed Image Text: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.
Expert Solution
Step 1

Let X,◊ and Y,* are two groups.

The Cartesian product of X and Y defined by X×Y=x,y | x∈X and  y∈Y.

Prove X×Y is a group under the operation x1,y1● x2,y2=x1◊y1, x2*y2 .

(i) Closure property

Since x1◊y1 and x2*y2 in X and Y, for all x1, y1 and x2, y2 in X×Y the binary operationx1, y1● x2, y2=x1◊y1, x2*y2  in X×Y.

(ii) Associative property

Let x1, y1, x2, y2 and x3, y3 in X×Y. Then prove that x1, y1●x2, y2●x3, y3=x1, y1●x2, y2●x3, y3.

Consider the left hand side of the above,

x1, y1●x2, y2●x3, y3=x1◊x2, y1*y2●x3,y3=x1◊x2◊x3, y1*y2*y3=x1, y1●x2◊x3, y2*y3=x1, y1●x2, y2●x3, y3

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