What does group mean in maths?

Answer 1

In mathematics, a group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. Specifically, for a set ( G ) and a binary operation ( * ), the group must satisfy: (1) closure, meaning for any ( a, b \in G ), the result of ( a * b ) is also in ( G ); (2) associativity, where ( (a * b) * c = a * (b * c) ); (3) an identity element ( e ) exists such that ( a * e = e * a = a ) for all ( a \in G ); and (4) every element ( a ) in ( G ) has an inverse ( b ) such that ( a * b = b * a = e ). Groups are foundational structures in abstract algebra and have applications across various areas of mathematics and science.