Finite group

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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RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

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

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

Some of these can be found at:

• Category: Variations of finiteness (groups)
• Category: Group properties tautological for finite groups
• Category: Group properties giving finiteness condition

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
• Also, the order of any quotient group divides the order of the group
• Sylow's theorem tells us that for any prime $p$, there exist $p$-Sylow subgroups, viz $p$-subgroups whose index is relatively prime to $p$.

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

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 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}, Page 2 (definition introduced in paragraph)
• A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, ^{More info}, Page 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}, Page 24 (definition introduced in paragraph, along with notion of order of a group)
• Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, ^{More info}, Page 56
• Topics in Algebra by I. N. Herstein, ^{More info}, Page 28 (definition introduced in paragraph)