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Which one of the following is NOT necessarily a property of a Group?

1. Commutativity
2. Associativity
3. Existence of inverse for every element
4. Existence of identity
edited | 915 views

Groupoid- closure property.

Semigroup- closure,associative.

Monoid-closure,associative,identity.

Group-closure,associative,identity,inverse.

Abelian group- group property+commutative So ans should be A..

edited

(A) Commutativity is a property of Abelian group

Rest 3 and closure property is needed for a group.

A group is a setG, 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

Closure

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

Associativity

For all ab 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, and thus one speaks of theidentity element.

Inverse element

For each a in G, there exists an element b in G, commonly denoted a−1 (or −a, if the operation is denoted "+"), 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 •

Abelian Group

An abelian group is a setA, together with an operation • that combines any two elements a and to form another element denoted a • b. The symbol • is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, (A, •), must satisfy five requirements known as the abelian group axioms:

Closure

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

Associativity

For all ab and c in A, the equation (a • b) • c = a • (b • c) holds.

Identity element

There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = aholds.

Inverse element

For each a in A, there exists an element b in A such that a • b = b • a = e, where e is the identity element.

Commutativity

For all ab in Aa • b = b • a.

A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group"

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