AbelianGroup< T > Class Template Reference

#include <Abelian.hpp>

Inheritance diagram for AbelianGroup< T >:

Public Member Functions
	AbelianGroup ()=default
	Default constructor (trivial group) More...
	AbelianGroup (const T &a)
bool	istrivial () const
	Returns 1 if group is trivial. More...
bool	iscyclic () const
	Returns 1 if group is cyclic. More...
auto	operator[] (int i) const
	Returns the i-th summand of the group (unmutable). If the i-th summand is Z then this returns 1 and if it's Z/n it returns n. More...
auto &	operator[] (int i)
	Returns the i-th summand of the group (mutable reference version). If the i-th summand is Z then this returns 1 and if it's Z/n it returns n. More...
int	number_of_summands () const
	Returns the number of direct summands of the group. More...
bool	operator== (const AbelianGroup< T > &G) const
	Compares for isomorphism. More...
bool	operator!= (const AbelianGroup< T > &G) const
	Compares for non-isomorphism. More...
AbelianGroup< T >	operator+ (const AbelianGroup< T > &G) const
	The direct sum of groups. No sorting! More...
int	order () const
	The order of the group. Returns 0 if it's infinite. More...
int	order (const T &element) const
	The order of an element in the group, provided as an array that expresses it as a linear combination of the generators. Returns 0 if it's infinite. More...
template<typename Scalar >
void	normalize (Eigen::Matrix< Scalar, -1, -1 > &linmap) const
	Normalize linear map whose range is this group (reduce n in Z/n to 0 etc.). More...
void	normalize (T &element) const
	Normalizes an element in the group to have minimal amount of signs amongst its multiples. More...
auto	span (const T &a, const std::vector< T > &b) const
	Expresses a as a linear combination of elements b[i] in the group. If a is not in their span, returns empty matrix. More...
bool	inSpan (const T &a, const std::vector< T > &b) const
	Finds if an element is in the span of other given elements in the group. More...
int	isMultiple (const T &a, const T &b) const
	Returns k if a=k*b in group. Returns 0 if a is not a multiple of b (so make sure a!=0) More...
std::pair< std::vector< dense_t< T > >, std::vector< dense_t< T > > >	all_automorphisms () const
	Returns all automorphisms of the group together with their inverses. Currently only for p-groups and Z+p-groups (otherwise returns empty) More...

Friends
template<typename S >
std::ostream &	operator<< (std::ostream &os, const AbelianGroup< S > &A)

Detailed Description

template<typename T>
class Mackey::AbelianGroup< T >

Class for finitely generated abelian groups

Constructor & Destructor Documentation

AbelianGroup() [1/2]

AbelianGroup ( )

default

Default constructor (trivial group)

AbelianGroup() [2/2]

AbelianGroup

(

const T &

a

)

Given an array of numbers a[0],a[1],...,a[n] constructs the group G(a[0])+G(a[1])+...+G(a[n]) where G(1)=Z and G(n)=Z/n if n>=1 We assume that in the provided array, a[i] divides a[j] if i<=j

Member Function Documentation

istrivial()

bool istrivial

(

)

const

Returns 1 if group is trivial.

iscyclic()

bool iscyclic

(

)

const

Returns 1 if group is cyclic.

operator[]() [1/2]

auto operator[]

(

int

i

)

const

Returns the i-th summand of the group (unmutable). If the i-th summand is Z then this returns 1 and if it's Z/n it returns n.

operator[]() [2/2]

auto& operator[]

(

int

i

)

Returns the i-th summand of the group (mutable reference version). If the i-th summand is Z then this returns 1 and if it's Z/n it returns n.

number_of_summands()

int number_of_summands

(

)

const

Returns the number of direct summands of the group.

operator==()

bool operator==

(

const AbelianGroup< T > &

G

)

const

Compares for isomorphism.

operator!=()

bool operator!=

(

const AbelianGroup< T > &

G

)

const

Compares for non-isomorphism.

operator+()

AbelianGroup<T> operator+

(

const AbelianGroup< T > &

G

)

const

The direct sum of groups. No sorting!

order() [1/2]

int order

(

)

const

The order of the group. Returns 0 if it's infinite.

order() [2/2]

int order

(

const T &

element

)

const

The order of an element in the group, provided as an array that expresses it as a linear combination of the generators. Returns 0 if it's infinite.

normalize() [1/2]

void normalize

(

Eigen::Matrix< Scalar, -1, -1 > &

linmap

)

const

Normalize linear map whose range is this group (reduce n in Z/n to 0 etc.).

normalize() [2/2]

void normalize

(

T &

element

)

const

Normalizes an element in the group to have minimal amount of signs amongst its multiples.

span()

auto span	(	const T &	a,
		const std::vector< T > &	b
	)		const

Expresses a as a linear combination of elements b[i] in the group. If a is not in their span, returns empty matrix.

inSpan()

bool inSpan	(	const T &	a,
		const std::vector< T > &	b
	)		const

Finds if an element is in the span of other given elements in the group.

isMultiple()

int isMultiple	(	const T &	a,
		const T &	b
	)		const

Returns k if a=k*b in group. Returns 0 if a is not a multiple of b (so make sure a!=0)

all_automorphisms()

std::pair<std::vector<dense_t<T> >, std::vector<dense_t<T> > > all_automorphisms

(

)

const

Returns all automorphisms of the group together with their inverses. Currently only for p-groups and Z+p-groups (otherwise returns empty)

Friends And Related Function Documentation

operator<<

std::ostream& operator<< ( std::ostream &  os, const AbelianGroup< S > &  A  )

friend

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The documentation for this class was generated from the following file:

• Abelian.hpp