Group theory and generalizations Definition
Order of a group
Builds on Group
Definition
Let be a group. The order of , written , is the cardinality of the underlying set of . When is finite, is the number of elements of . When is infinite, has infinite order.
Remarks
The trivial group has order , and the group of th roots of unity has order . The group has infinite order. Lagrange's theorem says that the order of a subgroup of a finite group divides the order of the group.
Reading path
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- Group concept
- Order of a group definition
Where this leads
42 pages build on this one.
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Order of a cyclic subgroup equals the order of its generator
- Order of a group is the index times the order of the subgroup
- Groups of prime order are cyclic
- Cyclic groups of the same order are isomorphic
- Classification of finite abelian groups
- Product of cyclic groups is cyclic iff the orders are coprime
- Order of an element divides the order of the group
- Fixed points of a p-group action
- Second Sylow theorem
- The number of Sylow subgroups is the index of the normaliser
- Order of the dihedral group
- Cyclic group of order n
- Abelian simple groups have prime order
- Cauchy's theorem
- Order of the general linear group over a finite field
- The number of Sylow subgroups divides the index
- A finite abelian group is cyclic iff its exponent equals its order
- Order of the symmetric group
- Third Sylow theorem
- The group of units of Z/pZ is cyclic
- Frattini argument
- Structure theorem for finitely generated abelian groups
- Size of an orbit times the size of the stabiliser
- Order of the group of units of Z/nZ is Euler's totient
- Order of the alternating group
- First Sylow theorem
- Groups of order pq
- Burnside's lemma
- g to the order of the group is the identity
- Galois group of a prime degree polynomial with two nonreal roots
- Hall's theorem on solvable groups
- Proof that the order of an element divides the order of the group
- Hall subgroup
- Sylow theorems
- Frattini subgroup
- Lattice-ordered group
- Proof that groups of prime order are cyclic
- Structure theorem for compactly generated LCA groups
- Left-orderable group
- Torsion-free group
- Orderable group
Further reading
- Web https://en.wikipedia.org/wiki/Order_(group_theory), lead section
About this page
| Tag | 0SEX9F9J5Z |
|---|---|
| URI | m2://0SEX9F9J5Z |
| Kind | definition |
| Writing | W1 stated |
| MSC | 20A05 |
| Defines | Order of a group |
| Uses | Group |
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Used by
- Order of a cyclic subgroup equals the order of its generator
- Order of a group is the index times the order of the subgroup
- Cyclic groups of the same order are isomorphic
- Order of the dihedral group
- Cyclic group of order n
- Order of the general linear group over a finite field
- Order of the symmetric group
- Order of the group of units of Z/nZ is Euler's totient