# 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 o

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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 is __________.

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Concept:

Inverse property:

• (a*b) = (b*a) = e(identity element) then b is called inverse of an element a, denoted by a-1

Group properties:

• In a group the identity element e of G is always unique
• Inverse of any element in G is unique
• Inverse of identity element e is e itself.

Explanation:

• x * x = e implies x is inverse of itself.
• y * y = e implies y is inverse of itself
• (x * y) *(x * y) = e implies x * y is inverse of itself
• (y * x) *(y * x) = e implies y*x is inverse of itself
• x * y = x * e * y= x * (x * y * x * y) * y = ( x * x )* y * x (y * y) = e * y * x * e = y * x
• x * y = y * x we can also write (x * y )*( y * x) =e  or (y * x) *(x* y) = e
•  x*y have two inverse x*y and y*x similarly y*x has two inverse y*x and x*y but according to group property every element has unique inverse so one will be considered in group not both.
• Maximum 4 elements is possible in this G are { e, x, y and x*y} or { e, x, y and y*x }.