COMMUTATIVE GROUP Meaning and
Definition
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A commutative group, also commonly known as an abelian group, is a mathematical structure consisting of a set of elements along with an operation that satisfies two properties: closure and commutativity.
Firstly, closure refers to the property that when two elements from the set are combined using the operation, the result is also an element of the set. In other words, for any elements a and b in the set, the operation (often denoted as +) applied to them, a + b, yields another element that is also part of the set.
Secondly, commutativity refers to the property that the order in which elements are combined does not affect the result. Specifically, for any two elements a and b in the set, the operation applied to them, a + b, gives the same result as b + a. This property is often denoted as a + b = b + a.
When a group satisfies both closure and commutativity, it is referred to as a commutative group. This means that any elements within the group can be combined in any order, and the result will be the same. Addition of integers or real numbers is a common example of a commutative group, as the sum of any two numbers is independent of their order. Other examples include the set of complex numbers under addition, and the set of rational numbers under multiplication.
Common Misspellings for COMMUTATIVE GROUP
- xommutative group
- vommutative group
- fommutative group
- dommutative group
- cimmutative group
- ckmmutative group
- clmmutative group
- cpmmutative group
- c0mmutative group
- c9mmutative group
- conmutative group
- cokmutative group
- cojmutative group
- comnutative group
- comkutative group
- comjutative group
- commytative group
- commhtative group
- commjtative group
Etymology of COMMUTATIVE GROUP
The term "commutative group" is derived from two distinct concepts: "commutative" and "group".
The adjective "commutative" originates from the Latin word "commutare", which means "to change or exchange". In mathematics, the commutative property refers to the ability to interchange order without affecting the outcome. For example, in the context of addition, it means that a + b = b + a for any values of a and b. Likewise, in multiplication, it implies that a * b = b * a.
The term "group" has its roots in the Latin word "gruppus", which means "cluster" or "bunch". The concept of a group in mathematics was coined by Arthur Cayley in the mid-19th century.
Similar spelling word for COMMUTATIVE GROUP
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