This article is about a basic definition in group theory. The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this: (facts closely related to Finite group, all facts related to Finite group) |Survey articles about this |
VIEW RELATED: Analogues of this | Variations of this | [SHOW MORE]
This article defines a group property that is pivotal (i.e., important) among existing group properties
View a list of pivotal group properties | View a complete list of group properties [SHOW MORE]
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.
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
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|subgroup-closed group property||Yes||Suppose is a finite group and is a subgroup of . Then, is also a finite group.|
|quotient-closed group property||Yes||Suppose is a finite group and is a normal subgroup of . Then, the quotient group is also a finite group.|
|finite direct product-closed group property||Yes||Suppose are finite groups. Then, the direct product 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 is finite if and only if its lattice of subgroups, is finite. Therefore if two groups 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
|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)||||
|odd-order group||finite group whose order is an odd number||(by definition)||||
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|||||--|
|finite abelian group||abelian group|||||||
|finite nilpotent group||nilpotent group|||||Finitely generated nilpotent group, Periodic nilpotent group|FULL LIST, MORE INFO|
|finite solvable group||solvable group|||||Finitely generated solvable group, Finitely presented solvable group, Polycyclic group, Solvable group generated by finitely many periodic elements|FULL LIST, MORE INFO|
|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|