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Finite group

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This article is about a basic definition in group theory.The article text may, however, contain advanced material.
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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

Symbol-free definition

A group is said to be finite if the cardinality of its underlying set (viz its order) is finite.

Definition with symbols

A group G is finite if the cardinality of the set G is finite.

Examples

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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 n. 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.


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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Conjunction with other properties

Weaker properties

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:

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.

References

Textbook references

Facts about Finite groupRDF feed
Defined inBook:DummitFoote (?, ?, ?)  +, Book:AlperinBell (?, ?, ?)  +, Book:Fraleigh (?, ?, ?)  +, Book:Hungerford (?, ?, ?)  +, Book:Gallian (?, ?, ?)  +, and Book:Herstein (?, ?, ?)  +
Page classTerm  +
Referenced inBook:DummitFoote (?, ?, ?)  +, Book:AlperinBell (?, ?, ?)  +, Book:Fraleigh (?, ?, ?)  +, Book:Hungerford (?, ?, ?)  +, Book:Gallian (?, ?, ?)  +, and Book:Herstein (?, ?, ?)  +
Stronger thanPeriodic group  +, Group of finite exponent  +, Finitely generated group  +, Locally finite group  +, and Slender group  +
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