Class SymmetricGroup

java.lang.Object

org.episteme.core.mathematics.algebra.groups.SymmetricGroup

All Implemented Interfaces:

Group<SymmetricGroup.Permutation>, Magma<SymmetricGroup.Permutation>, Monoid<SymmetricGroup.Permutation>, Set<SymmetricGroup.Permutation>

public class SymmetricGroup
extends Object
implements Group<SymmetricGroup.Permutation>

Represents the Symmetric Group S_n (group of all permutations of n elements).

The order of S_n is n!.

Since:

1.0

Author:

Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static class	SymmetricGroup.Permutation	Represents a permutation of {0, 1, ..., n-1}.

Constructor Summary

Constructors

Constructor	Description
SymmetricGroup(int n)

Method Summary

Modifier and Type	Method	Description
boolean	contains(SymmetricGroup.Permutation element)	Tests whether this set contains the specified element.
String	description()	Returns a human-readable description of this set.
SymmetricGroup.Permutation	identity()	Returns the identity element of this monoid.
SymmetricGroup.Permutation	inverse(SymmetricGroup.Permutation element)	Returns the inverse of the given element.
boolean	isCommutative()	Tests whether this is an abelian (commutative) group.
boolean	isEmpty()	Returns true if this set contains no elements.
SymmetricGroup.Permutation	operate(SymmetricGroup.Permutation left, SymmetricGroup.Permutation right)	Performs the binary operation on two elements.

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface Monoid

isAssociative

Constructor Details

SymmetricGroup

public SymmetricGroup(int n)

Method Details

operate

public SymmetricGroup.Permutation operate(SymmetricGroup.Permutation left,
 SymmetricGroup.Permutation right)

Description copied from interface: Magma

Performs the binary operation on two elements.

This is the fundamental operation of a magma. The result must be an element of this magma (closure property).

Properties: None required (not necessarily associative or commutative)

Specified by:

operate in interface Magma<SymmetricGroup.Permutation>

Parameters:

left - the first operand

right - the second operand

Returns:

the result of a ∗ b

See Also:

• Magma.isAssociative()
• Magma.isCommutative()

identity

public SymmetricGroup.Permutation identity()

Description copied from interface: Monoid

Returns the identity element of this monoid.

Satisfies: e ∗ a = a ∗ e = a for all a.

Specified by:

identity in interface Monoid<SymmetricGroup.Permutation>

Returns:

the identity element

inverse

public SymmetricGroup.Permutation inverse(SymmetricGroup.Permutation element)

Description copied from interface: Group

Returns the inverse of the given element.

For element a, returns a⁻¹ such that: a ∗ a⁻¹ = a⁻¹ ∗ a = e (identity).

Specified by:

inverse in interface Group<SymmetricGroup.Permutation>

Parameters:

element - the element to invert

Returns:

the inverse element

isCommutative

public boolean isCommutative()

Description copied from interface: Group

Tests whether this is an abelian (commutative) group.

Specified by:

isCommutative in interface Group<SymmetricGroup.Permutation>

Specified by:

isCommutative in interface Magma<SymmetricGroup.Permutation>

Returns:

true if this group is abelian

description

public String description()

Description copied from interface: Set

Returns a human-readable description of this set.

Examples:

• "ℝ (Real Numbers)"
• "ℤ/12ℤ (Integers modulo 12)"
• "{1, 2, 3, 4, 5}"

Specified by:

description in interface Set<SymmetricGroup.Permutation>

Returns:

a description of this set

isEmpty

public boolean isEmpty()

Description copied from interface: Set

Returns true if this set contains no elements.

The empty set (∅) is a fundamental concept in set theory. It is the unique set containing no elements.

Specified by:

isEmpty in interface Set<SymmetricGroup.Permutation>

Returns:

true if this set is empty

contains

public boolean contains(SymmetricGroup.Permutation element)

Description copied from interface: Set

Tests whether this set contains the specified element.

This is the fundamental operation of a set - membership testing.

Specified by:

contains in interface Set<SymmetricGroup.Permutation>

Parameters:

element - the element to test for membership

Returns:

true if this set contains the element, false otherwise

See Also:

• Set.isEmpty()
• invalid reference

  #size()