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
Symbol-free definition
A group is said to be finite if the cardinality of its underlying set (viz its order) is finite.
Definition with symbols
A group G is finite if the cardinality of the set G is finite.
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.
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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Conjunction with other properties
- Finite cyclic group is the conjunction with the property of being a cyclic group.
- Finite abelian group is the conjunction with the property of being an abelian group.
- Finite nilpotent group is the conjunction with the property of being a nilpotent group.
- Finite solvable group is the conjunction with the property of being a solvable group.
Weaker properties
- Periodic group: A group in which every element has finite order.
- Group of finite exponent
- Finitely generated group
- Locally finite group
- Slender group
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
- Also, the order of any quotient group divides the order of the 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.
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 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 0387945261, More info, Page 2 (definition introduced in paragraph)
- A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, More info, Page 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 0387905189, More info, Page 24 (definition introduced in paragraph, along with notion of order of a group)
- Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, More info, Page 56
- Topics in Algebra by I. N. Herstein, More info, Page 28 (definition introduced in paragraph)
| Defined in | Book:DummitFoote (?, ?, ?) +, Book:AlperinBell (?, ?, ?) +, Book:Fraleigh (?, ?, ?) +, Book:Hungerford (?, ?, ?) +, Book:Gallian (?, ?, ?) +, and Book:Herstein (?, ?, ?) + |
| Page class | Term + |
| Referenced in | Book:DummitFoote (?, ?, ?) +, Book:AlperinBell (?, ?, ?) +, Book:Fraleigh (?, ?, ?) +, Book:Hungerford (?, ?, ?) +, Book:Gallian (?, ?, ?) +, and Book:Herstein (?, ?, ?) + |
| Stronger than | Periodic group +, Group of finite exponent +, Finitely generated group +, Locally finite group +, and Slender group + |
