Group
A group is a set with a binary operation and an identity element satisfying closure, associativity, and inverse axioms.
Definition 1.2 (Group)
A group is a triple where
- is a set
- is a binary operation on ,
that satisfies the following four axioms:
-
Closure. For all , . [This can be deduced by definition of binary operations.]
-
Associativity. For all , .
-
Identity. For all , .
-
(Right) Inverse. For all , there exists such that .
Examples
The symmetries of an equilateral triangle form a group. The additive group is another example.
Stated in
- Definition 1.2 (Group)§1.2 Introduction on Groups