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- A group is a set of elements with a binary operation that satisfies four properties12345. The properties are:
- Closure: The result of the operation on any two elements in the set is also in the set12345.
- Associativity: The order of grouping the elements does not affect the result of the operation12345.
- Identity: There is an element in the set that does not change any other element when combined with the operation12345.
- Inverse: Every element in the set has another element that, when combined with the operation, gives the identity element12345.
Learn more:✕This summary was generated using AI based on multiple online sources. To view the original source information, use the "Learn more" links.A group is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property.mathworld.wolfram.com/Group.htmlProperties of Groups Every group (G, o) satisfies the following properties: The composition of two elements always belongs to G. That is, aob∈G for all a,b in G. Every group (G, 0) contains only one identity element. Each element in a group (G, o) has only one inverse. In a group (G, o), the cancellation law holds.www.mathstoon.com/group-theory/A group, G, is a finite or infinite set of components/factors, unitedly through a binary operation or group operation, that jointly meet the four primary properties of the group, i.e closure, associativity, the identity, and the inverse property.www.analyticssteps.com/blogs/what-group-theory-…A group is a set G G together with a binary operation ∘:G×G →G ∘: G × G → G with the following properties:
- Associativity. For any a,b,c∈G a, b, c ∈ G, we have that (a∘b)∘c =a∘(b∘c) ( a ∘ b) ∘ c = a ∘ ( b ∘ c).
algebrology.github.io/groups-and-their-basic-proper…Properties of Group Theory
- Conclusion: If A and B are two components in the G group, then "AB" is also in "G."
- Associativity: Since the defined multiplication is associative, if all A, B, and C belong to the same group, "G," then (AB) C = A. (BC).
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