Group order

The typeclass of groups, equipped with the function computing the order of a specific element of a group.

The order of x is the smallest positive integer k such that gtimes k x == mempty. If there are no such integers, the order of x is defined to be infinity.

Note: For any valid instances of GroupOrder, order x == Finite 1 holds if and only if x == mempty.

Examples:

>>> order (3 :: Sum Word8)
Finite 256
>>> order (16 :: Sum Word8)
Finite 16
>>> order (0 :: Sum Integer)
Finite 1
>>> order (1 :: Sum Integer)
Infinite

The order of a group element.

The order of a group element can either be infinite, as in the case of Sum Integer, or finite, as in the case of Sum Word8.

Unidirectional pattern synonym for the infinite order of a group element.

Unidirectional pattern synonym for the finite order of a group element.

An efficient implementation of order for additive group of fixed-width integers, like Int or Word8.

lcmOrder x y calculates the least common multiple of two Orders.

If both x and y are finite, it returns Finite r where r is the least common multiple of them. Otherwise, it returns Infinite.

Examples:

>>> lcmOrder (Finite 2) (Finite 5)
Finite 10
>>> lcmOrder (Finite 2) (Finite 10)
Finite 10
>>> lcmOrder (Finite 1) Infinite
Infinite

A FiniteGroup is a Group whose underlying set is finite. This is equivalently a group object in \( FinSet \).

Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups.

Calculate the exponent of a particular element in a finite group.

Examples:

>>> finiteOrder @(Sum Word8) 3
256