group (original) (raw)

Group.
A group is a pair (G,*), where G is a non-empty set and “*” is a binary operation on G, such that the following conditions hold:

If G is a group under *, then * is referred to as the group operation of G.

Usually, the symbol “*” is omitted and we write a⁢b fora*b. Sometimes, the symbol “+” is used to represent the operation, especially when the group is abelian.

It can be proved that there is only one identity element, and that for every element there is only one inverse. Because of this we usually denote the inverse of a as a-1 or -a when we are usingadditive notation. The identity element is also called neutral element due to its behavior with respect to the operation, and thusa-1 is sometimes (although uncommonly) called the neutralizing element of a. An element of a group besides the identity element is sometimes called a non-trivial element.