OrderOfElement

The order of a group element, the least n ≥ 1 with $g^n$ the identity, or Infinity when there is none.

WikipediaOrder (group theory)↗MathWorldGroupOrder↗mathlib4orderOf↗

OrderOfElement(g)The Monoid protocol's member for the order of an element, answered by whichever type g belongs to.

Typed signatures

Type	Package	Applies
(any) -> unknown	residuesreplaces structures	on IntegerMod, ResidueClass
(any) -> unknown	structures	always

Domain: Structures

Details

• The Monoid protocol's member: a type conforms by saying how it finds the order of one of its values. Not named Order, which is Wolfram's comparison Order[a, b].
• A permutation's order is the lcm of its cycle lengths: PermutationOrder's value.
• A unit of $\mathbb{Z}/m$, a ResidueClass, has MultiplicativeOrder's value. A non-unit has no order (no power of it reaches 1) and stays unevaluated.
• Among the exact numbers, only roots of unity have finite order: 1 has order 1, $-1$ has order 2, $i$ and $-i$ have order 4, and every other rational or Gaussian rational has infinite order. An inexact or symbolic number stays unevaluated.
• A matrix stays unevaluated: finite order is detectable by repeated multiplication, but infinite order is not, so none is guessed.

Examples

Basic #

the one-line word $312$ is a 3-cycle

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In$\mathrm{OrderOfElement}({\operatorname{Permutation}}([3, 1, 2]))$Out$3$

$3$ generates $(\mathbb{Z}/7)^\times$

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In$\mathrm{OrderOfElement}(\overline{3}_{7})$Out$6$

in the multiplicative group of the complex numbers

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In$\mathrm{OrderOfElement}(-1)$Out$2$

$i^4 = 1$

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In$\mathrm{OrderOfElement}(i)$Out$4$

Scope #

a 2-cycle and a 3-cycle: $\operatorname{lcm}(2,3)$

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In$\mathrm{OrderOfElement}({\operatorname{Permutation}}([2, 1, 4, 5, 3]))$Out$6$

only the identity has order 1

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In$\mathrm{OrderOfElement}({\operatorname{Permutation}}([1, 2, 3, 4]))$Out$1$

$5^2 = 25 \equiv 1 \pmod 8$

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In$\mathrm{OrderOfElement}(\overline{5}_{8})$Out$2$

the class of 1 has order 1

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In$\mathrm{OrderOfElement}(\overline{1}_{9})$Out$1$

$-1 \equiv 10 \pmod{11}$

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In$\mathrm{OrderOfElement}(\overline{-1}_{11})$Out$2$

$(-i)^4 = 1$

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In$\mathrm{OrderOfElement}(-i)$Out$4$

the identity

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In$\mathrm{OrderOfElement}(1)$Out$1$

Properties #

no power of $2$ is $1$

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In$\mathrm{OrderOfElement}(2)$Out$∞$

$(1/2)^n$ never reaches 1

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In$\mathrm{OrderOfElement}(\frac{1}{2})$Out$∞$

$(3+4i)/5$ has modulus 1 but is no root of unity

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In$\mathrm{OrderOfElement}(\frac{3}{5}+\frac{4}{5}i)$Out$∞$

the same value as PermutationOrder

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In$\mathrm{OrderOfElement}({\operatorname{Permutation}}([2, 4, 1, 3]))=\mathrm{PermutationOrder}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$\mathrm{True}$

the same value as MultiplicativeOrder

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In$\mathrm{OrderOfElement}(\overline{10}_{21})=\mathrm{MultiplicativeOrder}(10, 21)$Out$\mathrm{True}$

Possible issues #

$\gcd(10, 22) = 2$, so no power of $10$ is $1$ mod $22$

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In$\mathrm{OrderOfElement}(\overline{10}_{22})$Out$\mathrm{OrderOfElement}(\overline{10}_{22})$

this matrix has order 4, but a matrix is left unevaluated

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In$\mathrm{OrderOfElement}(\bigl\lbrack\bigl\lbrack0, -1\bigr\rbrack, \bigl\lbrack1, 0\bigr\rbrack\bigr\rbrack)$Out$\mathrm{OrderOfElement}(\bigl\lbrack\bigl\lbrack0, -1\bigr\rbrack, \bigl\lbrack1, 0\bigr\rbrack\bigr\rbrack)$

a float is not known to be exactly a root of unity

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In$\mathrm{OrderOfElement}(0.5)$Out$\mathrm{OrderOfElement}(0.5)$

nothing is known about x

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In$\mathrm{OrderOfElement}(x)$Out$\mathrm{OrderOfElement}(x)$

About

Examples	20
Crosswalk	WikipediaMathWorldmathlib4

See also: PermutationOrder, MultiplicativeOrder