In mathematics, a group is defined as a non-empty set G and a binary operation called the group operation.
As a shortcut is noted or even xy. This is called infix notation.
The group must obey the following rules (or axioms). Let a,b,c be arbitrary elements of G. Then:
An abelian group also obeys the additional rule:
Closure is part of the definition of "binary operation," so that A1 is often omitted.
The group operation can be any of a number of...
more
In mathematics, a group is defined as a non-empty set G and a binary operation called the group operation.
As a shortcut is noted or even xy. This is called infix notation.
The group must obey the following rules (or axioms). Let a,b,c be arbitrary elements of G. Then:
An abelian group also obeys the additional rule:
Closure is part of the definition of "binary operation," so that A1 is often omitted.
The group operation can be any of a number of operations. In generic terms, we denote the group operation, identity element and inverse of a: respectively.
There is also an additive notation, where those are replaced by + ,0, − a, and a multiplicative notation where those are replaced by * ,1,a .
The group is often referred to as "the group G" or simply "G"; but the operation "" is fundamental to the description of the group.
is usually pronounced "the group G under ". When affirming that G is a group (for example, in a theorem), we say that "G is a group under ".
A3 and A4 can...
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