# Finite group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Finite group, all facts related to Finite group) |Survey articles about this |

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This article defines a group property that is pivotal (i.e., important) among existing group properties

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## Definition

A group is said to be finite if the cardinality of its underlying set (i.e., its order) is finite. Here, the *cardinality* of a set refers to the *number of elements* in the set, and is denoted as .

## Examples

VIEW: groups satisfying this property | groups dissatisfying this propertyVIEW: |

The trivial group is an example of a finite group -- the underlying set has cardinality one. Other examples of finite groups include the symmetric group on a set, and the cyclic group of order . Any subgroup of a finite group is finite.

The group of integers, group of rational numbers, and group of real numbers (each under addition) are not finite groups.

## Facts

### Monoid generated is same as subgroup generated

In a finite group, the monoid generated by any subset is the same as the subgroup generated by it. This follows from the fact that since every element in a finite group has finite order, the inverse of any element can be written as a power of that element.

### Theorems on order-dividing

When we are working in finite groups, we can use results like these:

- Lagrange's theorem states that the order of any subgroup divides the order of the group
- order of element divides order of group
- order of quotient group divides order of group
- Sylow's theorem tells us that for any prime , there exist -Sylow subgroups, viz -subgroups whose index is relatively prime to .

### Existence of minimal and maximal elements

The lattice of subgroups of a finite group is a finite lattice, hence we can locate minimal elements and maximal elements, and do other things like find a finite stage at which every ascending/descending chain stabilizes.

Conversely, if the lattice of subgroups of a group is finite, then the group itself is finite. For a proof, see finitely many subgroups iff finite

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | Yes | Suppose is a finite group and is a subgroup of . Then, is also a finite group. | |

quotient-closed group property | Yes | Suppose is a finite group and is a normal subgroup of . Then, the quotient group is also a finite group. | |

finite direct product-closed group property | Yes | Suppose are finite groups. Then, the direct product is also a finite group. | |

lattice-determined group property | Yes | finitely many subgroups iff finite | Whether or not a group is finite is determined completely from its lattice of subgroups. Specifically, a group is finite if and only if its lattice of subgroups, is finite. Therefore if two groups have isomorphic lattices of subgroups, either both are finite, or neither is. |

## Relation with other properties

*This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied*

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### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

group of prime power order | finite group whose order is a power of a prime number | (by definition) | | | |

odd-order group | finite group whose order is an odd number | (by definition) | | |

### Conjunction with other properties

Conjunction | Other component of conjunction | Intermediate notions between finite group and conjunction | Intermediate notions between other component and conjunction |
---|---|---|---|

finite cyclic group | cyclic group | | | -- |

finite abelian group | abelian group | | | | |

finite nilpotent group | nilpotent group | | | Finitely generated nilpotent group, Periodic nilpotent group|FULL LIST, MORE INFO |

finite solvable group | solvable group | | | Finitely generated solvable group, Finitely presented solvable group, Polycyclic group, Solvable group generated by finitely many periodic elements|FULL LIST, MORE INFO |

### Weaker properties

## References

### Textbook references

Book | Page number | Chapter and section | Contextual information | View |
---|---|---|---|---|

Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info} |
17 | definition given as an additional comment after the formal definition of group | ||

Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261^{More info} |
2 | |||

A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907^{More info} |
58 | the term is not explicitly defined, but the definition is implicit in the section Finite groups and group tables |
||

Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189^{More info} |
24 | definition introduced in paragraph, along with notion of order of a group | ||

Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716^{More info} |
56 | |||

Topics in Algebra by I. N. Herstein^{More info} |
28 | definition introduced in paragraph |