Data.Group.Finite

Finite groups

class (Group g, Bounded g) => FiniteGroup g

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.

Instances

FiniteGroup ()

Defined in Data.Group.Finite

FiniteGroup a => FiniteGroup (Identity a)

Defined in Data.Group.Finite

FiniteGroup a => FiniteGroup (Dual a)

Defined in Data.Group.Finite

(Bounded a, Num a) => FiniteGroup (Sum a)

Defined in Data.Group.Finite

(FiniteGroup a, FiniteGroup b) => FiniteGroup (a, b)

Defined in Data.Group.Finite

FiniteGroup a => FiniteGroup (Proxy a)

Defined in Data.Group.Finite

(FiniteGroup a, FiniteGroup b, FiniteGroup c) => FiniteGroup (a, b, c)

Defined in Data.Group.Finite

FiniteGroup a => FiniteGroup (Const a b)

Defined in Data.Group.Finite

(FiniteGroup a, FiniteGroup b, FiniteGroup c, FiniteGroup d) => FiniteGroup (a, b, c, d)

Defined in Data.Group.Finite

(FiniteGroup a, FiniteGroup b, FiniteGroup c, FiniteGroup d, FiniteGroup e) => FiniteGroup (a, b, c, d, e)

Defined in Data.Group.Finite

Finite group combinators

finiteOrder :: (Eq g, FiniteGroup g) => g -> Natural

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

Examples:

>>> finiteOrder @(Sum Word8) 3
256

safeClassify :: (Eq a, Cyclic a, FiniteGroup a) => (a -> Bool) -> [a]

Classify elements of a finite Cyclic group.

Apply a classifying function a -> Bool to the elements of a Cyclic group as generated by its designated generator. This is a safer version of classify, that is gauranteed to terminate.

Examples:

>>> take 3 $ safeClassify (< (3 :: Sum Word8))
[Sum {getSum = 1},Sum {getSum = 2}]

Finite abelian groups

class FiniteGroup g => FiniteAbelianGroup g

Commutative FiniteGroups

Instances

FiniteAbelianGroup ()

Defined in Data.Group.Finite

FiniteAbelianGroup a => FiniteAbelianGroup (Identity a)

Defined in Data.Group.Finite

FiniteAbelianGroup a => FiniteAbelianGroup (Dual a)

Defined in Data.Group.Finite

(Num a, Bounded a) => FiniteAbelianGroup (Sum a)

Defined in Data.Group.Finite

(FiniteAbelianGroup a, FiniteAbelianGroup b) => FiniteAbelianGroup (a, b)

Defined in Data.Group.Finite

FiniteAbelianGroup a => FiniteAbelianGroup (Proxy a)

Defined in Data.Group.Finite

(FiniteAbelianGroup a, FiniteAbelianGroup b, FiniteAbelianGroup c) => FiniteAbelianGroup (a, b, c)

Defined in Data.Group.Finite

FiniteAbelianGroup a => FiniteAbelianGroup (Const a b)

Defined in Data.Group.Finite

(FiniteAbelianGroup a, FiniteAbelianGroup b, FiniteAbelianGroup c, FiniteAbelianGroup d) => FiniteAbelianGroup (a, b, c, d)

Defined in Data.Group.Finite

(FiniteAbelianGroup a, FiniteAbelianGroup b, FiniteAbelianGroup c, FiniteAbelianGroup d, FiniteAbelianGroup e) => FiniteAbelianGroup (a, b, c, d, e)

Defined in Data.Group.Finite