Group theory and generalizations Definition
Order of an element
Builds on GroupLaws of exponents in a group
Definition
Let be a group with identity element , and let . When there is a positive integer with , the order of is the smallest such . When there is no such , the element has infinite order. The notation for the order of is or .
Remarks
The identity element is the only element of order . In , each nonzero element has infinite order. In the multiplicative group , the element has order and the element has order . In a finite group, each element has finite order, and the order of equals the order of the subgroup that generates.
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
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- Group concept
- Laws of exponents in a group statement
- Order of an element definition
Where this leads
50 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
- Subgroups of a finite cyclic group correspond to divisors of its order
- Groups of prime order are cyclic
- P-group
- Cyclic groups of the same order are isomorphic
- Classification of finite abelian groups
- Finite p-groups are nilpotent
- 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
- Groups of order p squared are abelian
- The number of Sylow subgroups is the index of the normaliser
- Order of a permutation is the lcm of its cycle lengths
- g to the n equals 1 iff the order divides n
- Exponent of a group
- Automorphism group of a cyclic group
- Abelian simple groups have prime order
- Cauchy's theorem
- Sylow p-subgroup
- The number of Sylow subgroups divides the index
- A finite abelian group is cyclic iff its exponent equals its order
- Order of an inverse
- Third Sylow theorem
- Frattini argument
- Torsion subgroup
- Structure theorem for finitely generated abelian groups
- A nontrivial finite p-group has nontrivial centre
- First Sylow theorem
- Order of a power
- Number of elements of each order in a cyclic group
- Groups of order pq
- g to the order of the group is the identity
- Primitive polynomial over a finite field
- Galois group of a prime degree polynomial with two nonreal roots
- Orders of the elements of the integers modulo 6
- Hall's theorem on solvable groups
- Proof that the order of an element divides the order of the group
- The Klein four-group is abelian but not cyclic
- 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
- Orders of the elements of S3
- Orderable group
Further reading
- Web https://en.wikipedia.org/wiki/Order_(group_theory), lead section
About this page
| Tag | 0JPYSD5E0B |
|---|---|
| URI | m2://0JPYSD5E0B |
| Kind | definition |
| Writing | W1 stated |
| MSC | 20A05 |
| Defines | Order of an element |
| Uses | Group, Laws of exponents in a group |
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What this page needs for W2
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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What links here
Used by
- Order of a cyclic subgroup equals the order of its generator
- P-group
- Product of cyclic groups is cyclic iff the orders are coprime
- Order of a permutation is the lcm of its cycle lengths
- g to the n equals 1 iff the order divides n
- Exponent of a group
- Cauchy's theorem
- A finite abelian group is cyclic iff its exponent equals its order
- Order of an inverse
- Torsion subgroup
- Order of a power
- Primitive polynomial over a finite field
- Orders of the elements of the integers modulo 6
- The Klein four-group is abelian but not cyclic
- Orders of the elements of S3