Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian productX 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.Weneed 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)%3DNow we should show that (X × Y,•) is in fact a group.

Answered: Suppose now that we have two groups…

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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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 $(X \times Y) = \{(x, y) | x \in X a n d y \in Y\}$ .

Prove $(X \times Y)$ is a group under the operation $(x_{1}, y_{1}) ● (x_{2}, y_{2}) = (x_{1} ◊ y_{1}, x_{2} * y_{2})$ .

(i) Closure property

Since $x_{1} ◊ y_{1}$ and $x_{2} * y_{2}$ in X and Y, for all $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ in $(X \times Y)$ the binary operation $(x_{1}, y_{1}) ● (x_{2}, y_{2}) = (x_{1} ◊ y_{1}, x_{2} * y_{2})$ in $(X \times Y)$ .

(ii) Associative property

Let $(x_{1}, y_{1})$ , $(x_{2}, y_{2})$ and $(x_{3}, y_{3})$ in $(X \times Y)$ . Then prove that $((x_{1}, y_{1}) ● (x_{2}, y_{2})) ● (x_{3}, y_{3}) = (x_{1}, y_{1}) ● ((x_{2}, y_{2}) ● (x_{3}, y_{3}))$ .

Consider the left hand side of the above,

$\begin{array}{rcl} ((x_{1}, y_{1}) ● (x_{2}, y_{2})) ● (x_{3}, y_{3}) & = & (x_{1} ◊ x_{2}, y_{1} * y_{2}) ● (x_{3}, y_{3}) \\ = & (x_{1} ◊ (x_{2} ◊ x_{3}), y_{1} * (y_{2} * y_{3})) \\ = & (x_{1}, y_{1}) ● (x_{2} ◊ x_{3}, y_{2} * y_{3}) \\ = & (x_{1}, y_{1}) ● ((x_{2}, y_{2}) ● (x_{3}, y_{3})) \end{array}$

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