Finitely presented groups

FPGroup

Finitely presented group. Such groups can be constructed a factors of free groups, see free_group.

For a group G of type FPGroup, the elements in gens(G) satisfy the relators of the underlying presentation.

Functions that compute subgroups of G return groups of type SubFPGroup.

FPGroupElem

Element of a finitely presented group.

The generators of a finitely presented group are displayed as f1, f2, f3, etc., and every element of a finitely presented group is displayed as product of the generators.

SubFPGroup

Subgroup of a finitely presented group, a group that is defined by generators that are elements of a group G of type FPGroup.

Operations for computing subgroups of a group of type FPGroup or SubFPGroup, such as derived_subgroup and sylow_subgroup, return groups of type SubFPGroup.

Note that functions such as relators do not make sense for proper subgroups of a finitely presented group.

SubFPGroupElem

Element of a subgroup of a finitely presented group.

The elements are displayed in the same way as the elements of full finitely presented groups, see FPGroupElem.

free_group(n::Int, s::VarName = :f; eltype::Symbol = :letter) -> FPGroup
free_group(L::VarName... ; eltype::Symbol = :letter) -> FPGroup
free_group(varnames_specifiers... ; eltype::Symbol = :letter) -> FPGroup

Return a free group.

The first form returns a free group of rank n, where the generators are printed as "$s1", "$s2", ..., the default being f1, f2, ...

The second form returns a free group of rank n, where n is the length of L, and L consists of strings, symbols or characters giving the variable names.

In the final form, the argument list consists of a sequence of one or more of the following:

1. A vector L of variable names.
2. A pair of the form A => B, where A is a VarName (so a string, symbol or character) and B is a range or more generally an AbstractVector. Then length(B) generators are defined whose names derive from a combination of A and the respective element of B. For example :x => 1:3 defines three generators x[1], x[2], x[3].
3. A pair of the form A => C, where A is again a VarName, and C is a tuple of ranges or v. For example "a" => (1:2, 1:2) defines four generators a[1, 1], a[2, 1], a[1, 2], a[2, 2].

For the second and third type, optionally the A part can contain the placeholder # to modify where the indices are inserted. For example "a#" => (1:2, 1:2) defines four generators a11, a21, a12, a22.

Also, instead of a range, any vector can be used. For example "#" => ([:x,:y], [:A, :B]) defines four generators xA, yA, xB, yB.

In all variants, if the optional keyword argument eltype is given and has the value :syllable then each element in the free group is internally represented by a vector of syllables, whereas a representation by a vector of integers is chosen in the default case of eltype == :letter.

Examples

julia> F = free_group(:a, :b)
Free group of rank 2

julia> w = F[1]^3 * F[2]^F[1] * F[-2]^2
a^2*b*a*b^-2

Here we show some of the different ways to create a free group.

julia> gens(free_group(2))
2-element Vector{FPGroupElem}:
 f1
 f2

julia> gens(free_group(2, :a))
2-element Vector{FPGroupElem}:
 a1
 a2

julia> gens(free_group(:u, :v))
2-element Vector{FPGroupElem}:
 u
 v

julia> gens(free_group([:a, :b], "x" => 1:2, 'y' => (1:2, 1:2)))
8-element Vector{FPGroupElem}:
 a
 b
 x[1]
 x[2]
 y[1, 1]
 y[2, 1]
 y[1, 2]
 y[2, 2]

@free_group(args...)

Return the free group obtained from free_group(args...) and introduce its generators as Julia variables into the current scope.

Examples

julia> F = @free_group(:a, :b)
Free group of rank 2

julia> a^2*b*a*b^-2
a^2*b*a*b^-2

Note that the varname => vector syntax for specifying a vector or matrix or general array of variables behaves slightly differently compared to free_group, as the following example demonstrates.

julia> U1 = free_group("x" => 1:3); gens(U1)
3-element Vector{FPGroupElem}:
 x[1]
 x[2]
 x[3]

julia> U2 = @free_group("x" => 1:3); gens(U2)
3-element Vector{FPGroupElem}:
 x1
 x2
 x3

julia> (x2^x1)^-1
x1^-1*x2^-1*x1

full_group(G::T) where T <: Union{SubFPGroup, SubPcGroup}
full_group(G::T) where T <: Union{FPGroup, PcGroup}

Return F, emb where F is the full pc group of f.p. group of which G is a subgroup, and emb is an embedding of G into F.

Examples

julia> G = perfect_group(FPGroup, 60, 1);

julia> H = sylow_subgroup(G, 2)[1];

julia> full_group(H)[1] == G
true

julia> full_group(G)[1] == G
true

relators(G::FPGroup)

Return a vector of relators for the full finitely presented group G, i.e., elements $[w_1, w_2, \ldots, w_n]$ in $F =$ free_group(ngens(G)) such that G is isomorphic with $F/[w_1, w_2, \ldots, w_n]$.

Examples

julia> f = @free_group(:x, :y);

julia> q = quo(f, [x^2, y^2, comm(x, y)])[1];  relators(q)
3-element Vector{FPGroupElem}:
 x^2
 y^2
 x^-1*y^-1*x*y

length(g::Union{FPGroupElem, SubFPGroupElem})

Return the length of g as a word in terms of the generators of its parent or of the full group of its parent if g is an element of a free group, otherwise an exception is thrown.

Examples

julia> F = @free_group(:F1, :F2);

julia> length(F1*F2^-2)
3

julia> length(one(F))
0

julia> length(one(quo(F, [F1])[1]))
ERROR: ArgumentError: the element does not lie in a free group

map_word(g::Union{FPGroupElem, SubFPGroupElem}, genimgs::Vector; genimgs_inv::Vector = Vector(undef, length(genimgs)), init = nothing)
map_word(v::Vector{Union{Int, Pair{Int, Int}}}, genimgs::Vector; genimgs_inv::Vector = Vector(undef, length(genimgs)), init = nothing)

Return the product $R_1 R_2 \cdots R_n$ that is described by g or v, respectively.

If g is an element of a free group $G$, say, then the rank of $G$ must be equal to the length of genimgs, g is a product of the form $g_{i_1}^{e_1} g_{i_2}^{e_2} \cdots g_{i_n}^{e_n}$ where $g_i$ is the $i$-th generator of $G$ and the $e_i$ are nonzero integers, and $R_j =$ imgs[$i_j$]$^{e_j}$.

If g is an element of (a subgroup of) a finitely presented group then the result is defined as map_word applied to a representing element of the underlying free group of full_group(parent(g)). In particular, genimgs are interpreted as the images of the generators of this free group, not of gens(parent(g)).

If the first argument is a vector v of integers $k_i$ or pairs k_i => e_i, respectively, then the absolute values of the $k_i$ must be at most the length of genimgs, and $R_j =$ imgs[$|k_i|$]$^{\epsilon_i}$ where $\epsilon_i$ is the sign of $k_i$ (times $e_i$).

If a vector genimgs_inv is given then its assigned entries are expected to be the inverses of the corresponding entries in genimgs, and the function will use (and set) these entries in order to avoid calling inv (more than once) for entries of genimgs.

If init is different from nothing then the product gets initialized with init.

If v has length zero then init is returned if also genimgs has length zero, otherwise one(genimgs[1]) is returned. Thus the intended value for the empty word must be specified as init whenever it is possible that the elements in genimgs do not support one.

Examples

julia> F = @free_group(:F1, :F2);

julia> imgs = gens(symmetric_group(4))
2-element Vector{PermGroupElem}:
 (1,2,3,4)
 (1,2)

julia> map_word(F1^2, imgs)
(1,3)(2,4)

julia> map_word(F2, imgs)
(1,2)

julia> map_word(one(F), imgs)
()

julia> map_word(one(F), imgs, init = imgs[1])
(1,2,3,4)

julia> invs = Vector(undef, 2);

julia> map_word(F1^-2*F2, imgs, genimgs_inv = invs)
(1,3,2,4)

julia> invs
2-element Vector{Any}:
    (1,4,3,2)
 #undef