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Add accessors group and subgroup for SubgroupTransversal #4839

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Merged
fingolfin merged 7 commits into from
May 20, 2025
Merged

Add accessors group and subgroup for SubgroupTransversal #4839

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e5b7b18
Add `group` and `acting_group` for `SubgroupTransversal`
mjrodgers Apr 28, 2025
e938f44
Add `@ref` to `group`, `acting_group` in `SubgroupTransversal` docstring
mjrodgers Apr 28, 2025
dd9f40a
Add docstrings to main Documentation
mjrodgers Apr 28, 2025
c267fbd
improve docstring
mjrodgers Apr 28, 2025
eba7184
fix docstring
mjrodgers Apr 28, 2025
c2e19dd
rename `acting_group` to `subgroup`
mjrodgers May 9, 2025
9d3c70b
export `subgroup`
mjrodgers May 19, 2025
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3 changes: 3 additions & 0 deletions docs/src/Groups/subgroups.md
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Original file line number Diff line number Diff line change
Expand Up @@ -122,6 +122,9 @@ is_right(C::GroupCoset)
is_left(C::GroupCoset)
right_cosets(G::GAPGroup, H::GAPGroup; check::Bool=true)
left_cosets(G::GAPGroup, H::GAPGroup; check::Bool=true)
SubgroupTransversal
group(T::SubgroupTransversal)
subgroup(T::SubgroupTransversal)
right_transversal(G::T1, H::T2; check::Bool=true) where T1 <: GAPGroup where T2 <: GAPGroup
left_transversal(G::T1, H::T2; check::Bool=true) where T1 <: GAPGroup where T2 <: GAPGroup
is_bicoset(C::GroupCoset)
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57 changes: 57 additions & 0 deletions src/Groups/cosets.jl
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Expand Up @@ -373,9 +373,17 @@ end

Type of left/right transversals of subgroups in groups.

For a group $G$ and a subgroup $H$ of $G$, $T$ is a right
(resp. left) transversal for $H$ in $G$ if $T$ contains
precisely one element of each right (resp. left) cosets of $H$ in $G$.

Objects of this type are created by [`right_transversal`](@ref) and
[`left_transversal`](@ref).

- [`group(T::SubgroupTransversal)`](@ref) returns $G$.

- [`subgroup(T::SubgroupTransversal)`](@ref) returns $H$.

# Note for developers

The elements are encoded via a right transversal object in GAP.
Expand Down Expand Up @@ -436,6 +444,55 @@ function Base.keys(T::SubgroupTransversal)
return keys(1:length(T))
end

"""
group(T::SubgroupTransversal)

Return the group `G` that contains all of the elements in `T`.
That is, `T` is a left or right transversal of a subgroup of `G`.

# Examples
```jldoctest
julia> G = symmetric_group(5)
Sym(5)

julia> H = sylow_subgroup(G, 2)[1]
Permutation group of degree 5 and order 8

julia> T = right_transversal(G, H)
Right transversal of length 15 of
permutation group of degree 5 and order 8 in
Sym(5)

julia> group(T) == G
true
```
"""
group(T::SubgroupTransversal) = T.G

"""
subgroup(T::SubgroupTransversal)

Return the group `H` such that `T` is a (left or right)
transversal of `H`.
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@fingolfin fingolfin Apr 28, 2025

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But in how far is this group "acting"? If we look at the (right) cosets in $G/H = { Ha \mid a \in G }$ then I'd say this is a $G$-set, not an $H$-set (well, since $H$ is a subgroup, of course it would be valid to "restrict it" to $H$, but the most general starting point is to view it as a $G$-set.

Of course now we talk about a "transversal", which in general is neither a $G$-set nor an $H$-set. But one can in principle try to "fix" this, as follows: for `$r\in T$ and $g\in G$, define $r.g$ as the element $r' \in T$ for which $Hrg = Hr'$ (and at that point $rg/r'$ is a Schreier generator of $H$). This again suggests $G$ as acting group. Don't get me wrong: I am not saying we should implement such an action. Rather, I am trying to find with a rational for this naming, and coming up empty.

What am I missing?

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@mjrodgers mjrodgers Apr 28, 2025

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hm, I see your point. I wanted to use the same method names as for the individual cosets; but I agree, in this context G is the group that would be acting on the cosets.

I'm not really sure what would be a well-motivated way to name these methods (Magma doesn't have anything analogous, since they just treat a transversal as a generic ordered set)


# Examples
```jldoctest
julia> G = symmetric_group(5)
Sym(5)

julia> H = symmetric_group(3)
Sym(3)

julia> T = right_transversal(G, H)
Right transversal of length 20 of
Sym(3) in
Sym(5)

julia> subgroup(T) == H
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@fingolfin fingolfin May 13, 2025

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Doctests fail because subgroup is not exported

true
```
"""
subgroup(T::SubgroupTransversal) = T.H

"""
right_transversal(G::GAPGroup, H::GAPGroup; check::Bool=true)
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3 changes: 2 additions & 1 deletion src/exports.jl
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Expand Up @@ -1269,7 +1269,7 @@ export number_of_standard_tableaux
export number_of_transitive_groups, has_number_of_transitive_groups
export number_of_weak_compositions
export numerator
export numerical_lattice
export numerical_lattice
export numerical_lattice_of_K3_cover
export objective_function
export omega_group
Expand Down Expand Up @@ -1620,6 +1620,7 @@ export sub
export sub_object
export subalgebra_membership
export subalgebra_membership_homogeneous
export subgroup
export subgroup_classes
export subquo_type
export subquotient
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