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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
Q.
where e is the identity element. The maximum number of elements in such a group is __________.
where e is the identity element. The maximum number of elements in such a group is __________.
- a)2
- b)3
- c)4
- d)5
Correct answer is option 'C'. Can you explain this answer?
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There are two elements x, y in a group (G,∗) such that every el...
x * x = e, x is its own inverse
y * y = e, y 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
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}.
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.
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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 thatx ∗ x = y ∗ y = x ∗ y ∗ x ∗ y = y ∗ x ∗ y ∗ x = eQ.where e is the identity element. The maximum number of elements in such a group is __________.a)2b)3c)4d)5Correct answer is option 'C'. Can you explain this answer?
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