Real Analysis - Group

A group is a set, G, together with an operation "•" that combines any two elements a and b to form another element denoted a • b. The symbol "•" is a general placeholder for a concretely given operation, such as the addition above. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:

1. Closure. For all a, b in G, the result of the operation a • b is also in G.
2. Associativity. For all a, b and c in G, the equation (a • b) • c = a • (b • c) holds.
3. Identity element. There exists an element e in G, such that for all elements a in G, the equation e • a = a • e = a holds.
4. Inverse element. For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.