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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Contents
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
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VIEW: Related group property satisfactions | Related group property dissatisfactions
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 ![]() ![]() ![]() ![]() | |
quotient-closed group property | Yes | Suppose ![]() ![]() ![]() ![]() | |
finite direct product-closed group property | Yes | Suppose ![]() ![]() | |
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 ![]() ![]() ![]() |
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
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 |