GATE | GATE-CS-2014-(Set-3) | Question 59

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 __________.
(A) 2
(B) 3
(C) 4
(D) 5


Answer: (C)

Explanation:

x * x = e, x is its own inverse
y * y = e, y is its own inverse

(x*y) * (x* y) = e, x*y is its own inverse

(y*x) * (y*x) = e, y*x is its own inverse
also x*x*e = e*e can be rewritten as follows

x*y*y*x = e*y*y*e = e, (Since y *y = e)

(x*y) * (y*x) = e shows that (x *y) and (y *x)
are each other’s inverse and we already know that
(x*y) and (y*x) are inverse of its own.

As per (G,*) to be group any element should have
only one inverse element (unique)

This implies x*y = y*x (is one element)

So the elements of such group are 4 which are
{x, y, e, x*y}.

See following definition of group from wikipedia.

A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:[5]

Closure For all a, b in G, the result of the operation, a • b, is also in G.b[›]
Associativity For all a, b and c in G, (a • b) • c = a • (b • c).
Identity element There exists an element e in G, such that for every element a in G, the equation e • a = a • e = a holds. Such an element is unique (see below), and thus one speaks of the identity element.
Inverse element For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.

Source: https://en.wikipedia.org/wiki/Group_%28mathematics%29

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