Editing Group (mathematics) (section)

== Finite groups ==
{{Main|Finite group}}
A group is called ''finite'' if it has a [[finite set|finite number of elements]]. The number of elements is called the [[order of a group|order]] of the group.{{sfn|Kurzweil|Stellmacher|2004|loc=p. 3}} An important class is the ''[[symmetric group]]s'' {{tmath|1= \mathrm{S}_N }}, the groups of permutations of <math>N</math> objects. For example, the [[dihedral group of order 6|symmetric group on 3 letters]] <math>\mathrm{S}_3</math> is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 ([[factorial]] of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group <math>\mathrm{S}_N</math> for a suitable integer {{tmath|1= N }}, according to [[Cayley's theorem]]. Parallel to the group of symmetries of the square above, <math>\mathrm{S}_3</math> can also be interpreted as the group of symmetries of an [[equilateral triangle]].

The order of an element <math>a</math> in a group <math>G</math> is the least positive integer <math>n</math> such that {{tmath|1= a^n=e }}, where <math>a^n</math> represents
<math display=block>\underbrace{a \cdots a}_{n \text{ factors}},</math>
that is, application of the operation "{{tmath|1= \cdot }}" to <math>n</math> copies of {{tmath|1= a }}. (If "{{tmath|1= \cdot }}" represents multiplication, then <math>a^n</math> corresponds to the {{tmath|1= n }}th power of {{tmath|1= a }}.) In infinite groups, such an <math>n</math> may not exist, in which case the order of <math>a</math> is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.

More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups: [[Lagrange's theorem (group theory)|Lagrange's Theorem]] states that for a finite group <math>G</math> the order of any finite subgroup <math>H</math> [[divisor|divides]] the order of {{tmath|1= G }}. The [[Sylow theorems]] give a partial converse.

The dihedral group <math>\mathrm{D}_4</math> of symmetries of a square is a finite group of order 8. In this group, the order of <math>r_1</math> is 4, as is the order of the subgroup <math>R</math> that this element generates. The order of the reflection elements <math>f_{\mathrm{v}}</math> etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. The groups <math>\mathbb F_p^\times</math> of multiplication modulo a prime <math>p</math> have order {{tmath|1= p-1 }}.

=== Finite abelian groups ===
Any finite abelian group is isomorphic to a [[direct product|product]] of finite cyclic groups; this statement is part of the [[fundamental theorem of finitely generated abelian groups]].

Any group of prime order <math>p</math> is isomorphic to the cyclic group <math>\mathrm{Z}_p</math> (a consequence of [[Lagrange's theorem (group theory)|Lagrange's theorem]]). 
Any group of order <math>p^2</math> is abelian, isomorphic to <math>\mathrm{Z}_{p^2}</math> or {{tmath|1= \mathrm{Z}_p \times \mathrm{Z}_p }}.
But there exist nonabelian groups of order {{tmath|1= p^3 }}; the dihedral group <math>\mathrm{D}_4</math> of order <math>2^3</math> above is an example.<ref>{{harvnb|Artin|2018|loc=Proposition 6.4.3}}. See also {{harvnb|Lang|2002|p=77}} for similar results.</ref>

=== Simple groups ===
When a group <math>G</math> has a normal subgroup <math>N</math> other than <math>\{1\}</math> and <math>G</math> itself, questions about <math>G</math> can sometimes be reduced to questions about <math>N</math> and {{tmath|1= G/N }}. A nontrivial group is called ''[[simple group|simple]]'' if it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by the [[Jordan–Hölder theorem]].

=== Classification of finite simple groups ===
{{Main|Classification of finite simple groups}}

[[Computer algebra system]]s have been used to [[List of small groups|list all groups of order up to 2000]].{{efn|[[Up to]] isomorphism, there are about 49 billion groups of order up to 2000. See {{harvnb|Besche|Eick|O'Brien|2001}}.}}
But [[classification theorems|classifying]] all finite groups is a problem considered too hard to be solved.

The classification of all finite ''simple'' groups was a major achievement in contemporary group theory. There are [[List of finite simple groups|several infinite families]] of such groups, as well as 26 "[[sporadic groups]]" that do not belong to any of the families. The largest [[sporadic group]] is called the [[monster group]]. The [[monstrous moonshine]] conjectures, proved by [[Richard Borcherds]], relate the monster group to certain [[modular function]]s.{{sfn|Ronan|2007}}

The gap between the classification of simple groups and the classification of all groups lies in the [[extension problem]].<ref>{{harvnb|Aschbacher|2004|p=737}}.</ref>