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Question

# 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=y∗x∗y∗x=e where e is the identity element. The maximum number of elements in such a group is4

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Solution

## The correct option is A 4(i) e is identity element (ii) x∗x=e, so x=x−1 (iii) y∗y=e, so y=y−1 (iv) (x∗y)∗(x∗y)=e, so x∗y=(x∗y)−1 ……(i) and (y∗x)∗(y∗x)=e, so y∗x=(y∗x)−1 Now (x∗y)∗(y∗x)=x∗(y∗y)∗x=x∗e∗x =x∗x=e So, (x∗y)−1=(y∗x) ……(ii) From (i) and (ii) we get x∗y∗=y∗x There are only 4 distinct elements possible in this group 1. e 2. x 3. y 4. xy All other combinations are equal to one of these four as can be seen below: yx=xy (already proved) xxx=xe=x xyy=xe=x xxy=ey=y xyx=xxy=y and so on.... So the group is G={e,x,y,x∗y} ⇒|G|=4

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