CurrentModule = Oscar
DocTestSetup = Oscar.doctestsetup()
The following functions are available in OSCAR for subgroup properties:
sub(G::GAPGroup, gens::AbstractVector{<:GAPGroupElem}; check::Bool = true)
is_subset(H::GAPGroup, G::GAPGroup)
is_subgroup(H::GAPGroup, G::GAPGroup)
embedding(H::GAPGroup, G::GAPGroup)
index(G::GAPGroup, H::GAPGroup)
is_maximal_subgroup(H::GAPGroup, G::GAPGroup; check::Bool = true)
is_normalized_by(H::GAPGroup , G::GAPGroup)
is_normal_subgroup(H::GAPGroup, G::GAPGroup)
is_characteristic_subgroup(H::GAPGroup, G::GAPGroup; check::Bool = true)
The following functions are available in OSCAR to obtain standard subgroups of
a group G. Every such function returns a tuple (H,f), where H is a group
of the same type of G and f is the embedding homomorphism of H into G.
trivial_subgroup
center(G::GAPGroup)
sylow_subgroup(G::GAPGroup, p::IntegerUnion)
derived_subgroup
fitting_subgroup
frattini_subgroup
socle
solvable_radical
pcore(G::GAPGroup, p::IntegerUnion)
intersect(::T, V::T...) where T<:GAPGroup
The following functions return a vector of subgroups.
normal_subgroups
maximal_normal_subgroups
minimal_normal_subgroups
characteristic_subgroups
derived_series
sylow_system
hall_system
complement_system
chief_series
composition_series
jennings_series
p_central_series
lower_central_series
upper_central_series
!!! note
When a function returns a vector of subgroups,
the output consists in the subgroups only;
the embeddings are not returned as well.
To get the embedding homomorphism of the subgroup H in G,
one can type embedding(H, G).
The following functions return an iterator of subgroups. Usually it is more efficient to work with (representatives of) the underlying conjugacy classes of subgroups instead.
complements(G::GAPGroup, N::GAPGroup)
hall_subgroups
low_index_subgroups
maximal_subgroups
subgroups(G::GAPGroup)
is_conjugate(G::GAPGroup, x::GAPGroupElem, y::GAPGroupElem)
is_conjugate(G::GAPGroup, H::GAPGroup, K::GAPGroup)
is_conjugate_with_data(G::GAPGroup, x::GAPGroupElem, y::GAPGroupElem)
is_conjugate_with_data(G::GAPGroup, H::GAPGroup, K::GAPGroup)
centralizer(G::GAPGroup, x::GAPGroupElem)
centralizer(G::GAPGroup, H::GAPGroup)
normalizer(G::GAPGroup, x::GAPGroupElem)
normalizer(G::GAPGroup, H::GAPGroup)
core(G::GAPGroup, H::GAPGroup)
normal_closure(G::GAPGroup, H::GAPGroup)
GroupConjClass{T<:GAPGroup, S<:Union{GAPGroupElem,GAPGroup}}
representative(G::GroupConjClass)
acting_group(G::GroupConjClass)
number_of_conjugacy_classes(G::GAPGroup)
conjugacy_class(G::GAPGroup, g::GAPGroupElem)
conjugacy_class(G::GAPGroup, H::GAPGroup)
conjugacy_classes(G::GAPGroup)
complement_classes
hall_subgroup_classes
low_index_subgroup_classes
maximal_subgroup_classes(G::GAPGroup)
subgroup_classes(G::GAPGroup)
GroupCoset
right_coset(H::GAPGroup, g::GAPGroupElem)
left_coset(H::GAPGroup, g::GAPGroupElem)
is_right(c::GroupCoset)
is_left(c::GroupCoset)
is_bicoset(C::GroupCoset)
acting_domain(C::GroupCoset)
representative(C::GroupCoset)
right_cosets(G::GAPGroup, H::GAPGroup; check::Bool=true)
left_cosets(G::GAPGroup, H::GAPGroup; check::Bool=true)
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
GroupDoubleCoset{T <: GAPGroup, S <: GAPGroupElem}
double_coset(G::GAPGroup, g::GAPGroupElem, H::GAPGroup)
double_cosets(G::T, H::GAPGroup, K::GAPGroup; check::Bool=true) where T <: GAPGroup
left_acting_group(C::GroupDoubleCoset)
right_acting_group(C::GroupDoubleCoset)
representative(C::GroupDoubleCoset)
order(C::Union{GroupCoset,GroupDoubleCoset})
Base.rand(C::Union{GroupCoset,GroupDoubleCoset})
intersect(V::AbstractVector{Union{<: GAPGroup, GroupCoset, GroupDoubleCoset}})