GATE | GATE-CS-2009 | Question 1

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

Answer: (A)

Explanation: 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:

For all a, b in G, the result of the operation, a • b, is also in G.b

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.
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 • b = b • 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 • b = b • a always holds are called abelian groups (in honor of Niels Abel)


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