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# GATE | GATE-CS-2009 | Question 1

• Last Updated : 28 Jun, 2021

Which one of the following in NOT necessarily a property of a Group?
(A) Commutativity
(B) Associativity
(C) Existence of inverse for every element
(D) Existence of identity

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

Closure
For all a, b in G, the result of the operation, a â€¢ b, is also in G.b

Associativity
For all a, b and c in G, (a â€¢ b) â€¢ c = a â€¢ (b â€¢ c).

Identity element
There exists an element e in G, such that for every element a in G, the equation e â€¢ a = a â€¢ 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 â€¢ b = b â€¢ a = e, where e is the identity element.
The result of an operation may depend on the order of the operands. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation

a â€¢ b = b â€¢ a
may not always be true. This equation always holds in the group of integers under addition, because a + b = b + a for any two integers (commutativity of addition). Groups for which the commutativity equation a â€¢ b = b â€¢ a always holds are called abelian groups (in honor of Niels Abel)

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