###### Question 8144 – Theory-of-Computation

June 2, 2024###### Question 10322 – Computer-Organization

June 3, 2024# Question 8731 – Engineering-Mathematics

There are two elements x, y in a group (G,∗) such that every element in the group can be written as a product of some number of x’s and y’s in some order. It is known that

x * x = y * y = x * y * x = y * x * y * x = e

where e is the identity element. The maximum number of elements in such a group is ________.

Correct Answer: A

Question 126 Explanation:

We know

a*a

1. x*x = e So x

2. y*y = e So y

4. (y*x)*(y*x) = x*y*y*x = x*x*e = e So (y*x)

In ③, ④

x*y, y*x has same inverse, there should be unique inverse for each element.

x*y = y*x (even with cumulative law, we can conclude)

So {x, y, e, x*y} are element of Group.

a*a

^{-1}= e1. x*x = e So x

^{-1}is x ⇒ x is element of Group2. y*y = e So y

^{-1}= y ⇒ y is element of Group4. (y*x)*(y*x) = x*y*y*x = x*x*e = e So (y*x)

^{-1}= (y*x)In ③, ④

x*y, y*x has same inverse, there should be unique inverse for each element.

x*y = y*x (even with cumulative law, we can conclude)

So {x, y, e, x*y} are element of Group.

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