Definition of a Group A group is a non-empty set G together with an operation (or binary operation) on G that satisfies specific properties. Key properties...
A group is a non-empty set G together with an operation (or binary operation) on G that satisfies specific properties.
Key properties of a group:
Closure: For all elements a and b in G, if a + b is in G, then a - b is also in G.
Closure under the operation: The operation must be closed on G, meaning that for all elements a and b in G, if a + b is in G, then (a - b) is also in G.
Identity element: There exists an element e in G such that for all a in G, e + a = a and a + e = a.
Inverse elements: For each element a in G, there exists an element b in G such that a - b = e, where e is the identity element.
Associativity: The operation must be associative, meaning that for all elements a, b, and c in G, we have (a + b) + c = a + (b + c).
Examples:
The set of all integers under addition with the operation of addition is a group.
The set of all symmetries of a square is a group under the operation of composition.
The set of all permutations of a set of n elements is a group under the operation of composition.
Additional Notes:
A group with more than two elements is called a group.
A group with a single identity element is called a abelian group.
A group where the operation is commutative (a + b = b + a) is called a commutative group