Editing Group (mathematics) (section)

=== Numbers ===
Many number systems, such as the integers and the [[rational number|rationals]], enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as [[ring (mathematics)|rings]] and fields. Further abstract algebraic concepts such as [[module (mathematics)|module]]s, [[vector space]]s and [[algebra over a field|algebras]] also form groups.

==== Integers ====
The group of integers <math>\Z</math> under addition, denoted {{tmath|1= \left(\Z,+\right) }}, has been described above. The integers, with the operation of multiplication instead of addition, <math>\left(\Z,\cdot\right)</math> do ''not'' form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, <math>a=2</math> is an integer, but the only solution to the equation <math>a\cdot b=1</math> in this case is {{tmath|1= b=\tfrac{1}{2} }}, which is a rational number, but not an integer. Hence not every element of <math>\Z</math> has a (multiplicative) inverse.{{efn|Elements which do have multiplicative inverses are called [[unit (ring theory)|units]], see {{harvnb|Lang|2002|loc=§II.1|p=84}}.}}

==== Rationals ====
The desire for the existence of multiplicative inverses suggests considering [[fraction (mathematics)|fractions]]
<math display=block alt="a/b">\frac{a}{b}.</math>

Fractions of integers (with <math>b</math> nonzero) are known as [[rational number]]s.{{efn|The transition from the integers to the rationals by including fractions is generalized by the [[field of fractions]].}} The set of all such irreducible fractions is commonly denoted {{tmath|1= \Q }}. There is still a minor obstacle for {{tmath|1= \left(\Q,\cdot\right) }}, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no <math>x</math> such that {{tmath|1= x\cdot 0=1 }}), <math>\left(\Q,\cdot\right)</math> is still not a group.

However, the set of all ''nonzero'' rational numbers <math>\Q\smallsetminus\left\{0\right\}=\left\{q\in\Q\mid q\neq 0\right\}</math> does form an abelian group under multiplication, also denoted {{tmath|1= \Q^{\times} }}.{{efn|The same is true for any [[field (mathematics)|field]] <!--use {{math}}, since <math> in footnotes is unreadable on mobile devices-->{{math|''F''}} instead of {{math|'''Q'''}}. See {{harvnb|Lang|2005|loc=§III.1|p=86}}.}} Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of <math>a/b</math> is {{tmath|1= b/a }}, therefore the axiom of the inverse element is satisfied.

The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if [[division (mathematics)|division]] by other than zero is possible, such as in <math>\Q</math> – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.{{efn|For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See {{harvnb|Lang|2002|loc=Theorem IV.1.9}}. The notions of [[Torsion (algebra)|torsion]] of a [[module (mathematics)|module]] and [[simple algebra]]s are other instances of this principle.}}