Group

AbstractGroup

Abstract type for abstract groups. Currently the only subtype is FiniteGroup.

FiniteGroup

Finite group, with elements {1, 2, 3,..., n}. The identity element is always 1. Can be constructed using FiniteGroup(multiplication_table)

Fields

• multiplication_table::Matrix{Int}: multiplication table
• period_lengths::Vector{Int}: period length (order) of every element
• inverses::Vector{Int}: inverse of every element
• conjugacy_classes::Vector{Vector{Int}}: conjugacy classes

Examples

julia> using LatticeTools

julia> FiniteGroup([1 2; 2 1])
FiniteGroup([1 2; 2 1], [1, 2], [1, 2], [[1], [2]])

element(group, idx)

Return the element of index idx. For FiniteGroup, this is somewhat meaningless since the idxth element is idx. The sole purpose of this function is the bounds checking.

elements(group)

Return the elements of the group.

element_name(group, idx)

Return the name of element at index idx, which is just the string of idx.

element_names(group)

Return the names of element.

group_order(group)

Order of group (i.e. number of elements)

group_order(group, g)

Order of group element (i.e. period length)

period_length(group, g)

Order of group element (i.e. period length)

group_multiplication_table(group)

Return multiplcation table of the group.

isabelian(group)

Check if the group is abelian.

group_product(group)

Return a function which computes the group product.

group_inverse(group)

Get a function which gives inverse.

group_inverse(group, g)

Get inverse of element/elements g.

conjugacy_class(group::FiniteGroup, i::Integer)

Conjugacy class of the element i.

generate_subgroup(group::FiniteGroup, idx::Integer)

subgroup generated by generators. ⟨ {idx} ⟩

issubgroup(group, subset)

Check whether the given subset is a subgroup of group.

minimal_generating_set(group)

Get minimally generating set of the finite group.

group_isomorphism(group1, group2)

Find the isomorphism ϕ: G₁ → G₂. Return nothing if not isomorphic.

group_multiplication_table(elements, product=(*))

Generate a multiplication table from elements with product.

ishomomorphic(group, representation; product=(*), equal=(==))

Check whether representation is homomorphic to group under product and equal, order preserved.

Permutation(perms; max_order=2048)

Create a permutation of integers from 1 to n. perms should be a permutation of 1:n.

Arguments

• perms: an integer vector containing a permutation of integers from 1 to n
• max_order: maximum order

Note

The convention for the permutation is that map[i] gets mapped to i. In other words, map tells you where each element is from.

*(p1 ::Permutation, p2 ::Permutation)

Multiply the two permutation. NOT THIS: (Return [p2.map[x] for x in p1.map].) BUT THIS: (Return [p1.map[x] for x in p2.map].)

Examples

julia> using LatticeTools

julia> Permutation([2,1,3]) * Permutation([1,3,2])
Permutation([2, 3, 1], 3)

julia> Permutation([1,3,2]) * Permutation([2,1,3])
Permutation([3, 1, 2], 3)

generate_group(generators...)

Return a FiniteGroup generated by the generators.

Arguments

• generators::Permutation...: generating permutations

isidentity(perm::Permutation)

Test whether the permutation is an identity.