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 with symbols
A group is finite if the cardinality of the set is finite. In other words, has only finitely many elements.
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.
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|
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.
|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|