1.4k views

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 | 1.4k views

• Groupoid: closure property.
• Semigroup: closure, associative.
• Monoid-closure: associative, identity.
• Group: closure, associative, identity, inverse.
• Abelian group: group properties + commutativity

So, ans should be A.

by Active (2.4k points)
edited
0

A groupoid can be seen as a Group with a partial function replacing the binary operation.

Groupoid satisfies all properties of group not only closure property.

https://en.wikipedia.org/wiki/Groupoid

0

Ref :https://en.wikipedia.org/wiki/Groupoid (A) Commutativity is a property of Abelian group

Rest 3 and closure property is needed for a group.

by Active (2.4k points)

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" by Active (4.2k points)
0
wat is magma in the pic ???
0
magma is groupoid mam