271 Group
A set with an associative binary operation that has an identity element and inverses that is used to describe symmetry of equations.
definition (Group) A tuple \((G, *)\) to which these axioms apply:
- (Closure) \(a * b \in S\)
- (Identity) \(a * i = i * a = a\)
- (Inverse) \(a * a^{-1} = a^{-1} * a = i\)
- (Associativity) \(a*(b*c) = (a*b)*c\)
where
- \(G\) is the underlying set of the group.
- \(*\) is the binary operation on \(G\).
- \(a, b, c \in G\) are elements of the group.
- \(i\) is the identity element.
- \(a^{-1}\) is the inverse of \(a\).