Editing Group (mathematics) (section)

=== Notation and terminology ===
Formally, a group is an [[ordered pair]] of a set and a binary operation on this set that satisfies the [[group axioms]]. The set is called the ''underlying set'' of the group, and the operation is called the ''group operation'' or the ''group law''.

A group and its underlying set are thus two different [[mathematical object]]s. To avoid cumbersome notation, it is common to [[abuse of notation|abuse notation]] by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.

For example, consider the set of [[real number]]s {{tmath|1= \R }}, which has the operations of addition <math>a+b</math> and [[multiplication]] {{tmath|1= ab }}. Formally, <math>\R</math> is a set, <math>(\R,+)</math> is a group, and <math>(\R,+,\cdot)</math> is a [[field (mathematics)|field]]. But it is common to write <math>\R</math> to denote any of these three objects.

The ''additive group'' of the field <math>\R</math> is the group whose underlying set is <math>\R</math> and whose operation is addition. The ''multiplicative group'' of the field <math>\R</math> is the group <math>\R^{\times}</math> whose underlying set is the set of nonzero real numbers <math>\R \smallsetminus \{0\}</math> and whose operation is multiplication.

More generally, one speaks of an ''additive group'' whenever the group operation is notated as addition; in this case, the identity is typically denoted {{tmath|1= 0 }}, and the inverse of an element <math>x</math> is denoted {{tmath|1= -x }}. Similarly, one speaks of a ''multiplicative group'' whenever the group operation is notated as multiplication; in this case, the identity is typically denoted {{tmath|1= 1 }}, and the inverse of an element <math>x</math> is denoted {{tmath|1= x^{-1} }}. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, <math>ab</math> instead of {{tmath|1= a\cdot b }}.

The definition of a group does not require that <math>a\cdot b=b\cdot a</math> for all elements <math>a</math> and <math>b</math> in {{tmath|1= G }}. If this additional condition holds, then the operation is said to be [[commutative]], and the group is called an [[abelian group]]. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.

Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are [[function (mathematics)|functions]], the operation is often [[function composition]] {{tmath|1= f\circ g }}; then the identity may be denoted id. In the more specific cases of [[geometric transformation]] groups, [[symmetry (mathematics)|symmetry]] groups, [[permutation group]]s, and [[automorphism group]]s, the symbol <math>\circ</math> is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.