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

A group G 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 |G|.

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.


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.

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 G is a finite group and H is a subgroup of G. Then, H is also a finite group.
quotient-closed group property Yes Suppose G is a finite group and H is a normal subgroup of G. Then, the quotient group G/H is also a finite group.
finite direct product-closed group property Yes Suppose G_1, G_2, \dots, G_n are finite groups. Then, the direct product G_1 \times G_2 \times \dots \times G_n 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 G is finite if and only if its lattice of subgroups, L(G) is finite. Therefore if two groups G_1, G_2 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) |FULL LIST, MORE INFO
odd-order group finite group whose order is an odd number (by definition) |FULL LIST, MORE INFO

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

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
periodic group every element has finite order order of element divides order of group periodic not implies finite Artinian group, Finitely generated periodic group, Finitely presented periodic group, Group of finite exponent, Locally finite group|FULL LIST, MORE INFO
group of finite exponent exponent is finite, i.e., every element has finite order and the lcm of all the orders is finite order of element divides order of group finite exponent not implies finite |FULL LIST, MORE INFO
finitely generated group has a finite generating set finite implies finitely generated finitely generated not implies finite Finitely generated Hopfian group, Finitely generated group for which all homomorphisms to any finite group can be listed in finite time, Finitely generated periodic group, Finitely generated profinite group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Group with a finite complete rewriting system, Noetherian group|FULL LIST, MORE INFO
locally finite group every finitely generated subgroup is finite (by definition) locally finite not implies finite Group embeddable in a finitary symmetric group|FULL LIST, MORE INFO
Noetherian group every subgroup is finitely generated (by definition) Noetherian not implies finite Group of finite max-length|FULL LIST, MORE INFO

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 978-0471433347More 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 0387945261More info 2
A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907More 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 0387905189More info 24 definition introduced in paragraph, along with notion of order of a group
Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716More info 56
Topics in Algebra by I. N. HersteinMore info 28 definition introduced in paragraph