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: Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.

Contents

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:

• 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 $p$, 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.

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

View:

• Variations of finite group

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