[[Group theory MOC]]
# Group order

The **order** $\abs{G}$ of a [[group]] $G$ is the number of elements in that group. #m/def/group 
Similarly **order** $\abs a$ of an element $a \in G$ is the smallest integer $n$ such that $a^n=e$, #m/def/group where $a$ is said to have infinite order if no such $n$ exists.
The reason for this dual naming and notation is [[the order of a cyclic group equals the order of its generator]].

## Properties

- [[Lagrange's theorem|The order of a subgroup divides the order of a group]]
- [[The order of an element and its inverse are the same]]
- [[𝑎𝑏 and 𝑏𝑎 have the same order]]
- Elements of order greater than two come in pairs ($a, a^{-1}$)
- The product of two finite-order elements may be infinite-ordered[^ex]
- [[Number of elements of order 𝑑 in a finite group]]
- [[Order of powers of a group element]]
- [[Relationship between I𝑎𝑏I and I𝑎I𝑏I]]
- [[Cauchy's order theorem]]

[^ex]: See [[@gallianContemporaryAbstractAlgebra2017|Gallian]] §3 exercise 50

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