Axioms of a Group
(1) Closure(2)Associativity(3)Identity(4)Inverse
A group is defined as a set of elements that are closed under a binary operation and satisfy four axioms.
1. Closure: For any two elements a and b in the group, their product, denoted by ab, is also an element of the group.
2. Associativity: For any three elements a, b, and c in the group, the order of the multiplication doesn’t matter: (ab)c = a(bc).
3. Identity: There exists an element e in the group such that for any element a, ae = ea = a.
4. Inverse: For every element a in the group, there exists an element a⁻¹ in the group such that aa⁻¹ = a⁻¹a = e.
The closure axiom states that the product of any two elements in the group must be an element of the group. This ensures that the group is closed under the binary operation.
The associativity axiom states that the order of the multiplication of three or more elements does not matter, as long as the operands are the same. This ensures that the group is well-defined and the multiplication operation is associative.
The identity axiom states that there exists an element e in the group that does not affect any other element in the group when multiplied. This ensures that there is a starting point for any given multiplication operation.
The inverse axiom states that for every element in the group, there exists an inverse element that when multiplied with the element produces the identity element e. This ensures that there is a way to ‘undo’ any given multiplication operation.
These axioms are the building blocks of the theory of groups, which can be applied to algebra, geometry, and physics, among other areas.
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