address comments

docs/src/Groups/subgroups.md

@@ -67,6 +67,18 @@ upper_central_series
     one can type `embedding(G,H)`.
 
 
+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.
+
+```@docs
+complements(G::T, N::T) where T <: GAPGroup
+hall_subgroups
+low_index_subgroups
+maximal_subgroups
+subgroups(G::GAPGroup)
+```
+
 ## Conjugation action of elements and subgroups
 
 ```@docs

src/Groups/GAPGroups.jl

@@ -788,6 +788,23 @@ function subgroup_classes(G::GAPGroup; order::T = ZZRingElem(-1)) where T <: Int
   return [GAPGroupConjClass(G, _as_subgroup_bare(G, GAPWrap.Representative(cc)), cc) for cc in L]
 end
 
+"""
+    subgroups(G::GAPGroup)
+
+Return an iterator of all subgroups in `G`.
+Very likely it is better to use [`subgroup_classes`](@ref) instead.
+
+# Examples
+```jldoctest
+julia> println([order(H) for H in subgroups(symmetric_group(3))])
+ZZRingElem[1, 2, 2, 2, 3, 6]
+
+julia> println([order(H) for H in subgroups(quaternion_group(8))])
+ZZRingElem[1, 2, 4, 4, 4, 8]
+```
+"""
+subgroups(G::GAPGroup) = Iterators.flatten(subgroup_classes(G))
+
 """
     maximal_subgroup_classes(G::Group)
 
@@ -810,6 +827,23 @@ julia> maximal_subgroup_classes(G)
   return Vector{GAPGroupConjClass{T, T}}(LL)
 end
 
+"""
+    maximal_subgroups(G::Group)
+
+Return an iterator of the maximal subgroups in `G`.
+Very likely it is better to use [`maximal_subgroup_classes`](@ref) instead.
+
+# Examples
+```jldoctest
+julia> println([order(H) for H in maximal_subgroups(symmetric_group(3))])
+ZZRingElem[3, 2, 2, 2]
+
+julia> println([order(H) for H in maximal_subgroups(quaternion_group(8))])
+ZZRingElem[4, 4, 4]
+```
+"""
+maximal_subgroups(G::T) where T <: Union{GAPGroup, FinGenAbGroup} = Iterators.flatten(maximal_subgroup_classes(G))
+
 """
     low_index_subgroup_classes(G::GAPGroup, n::Int)
 
@@ -832,6 +866,22 @@ function low_index_subgroup_classes(G::GAPGroup, n::Int)
   return [conjugacy_class(G, H) for H in _as_subgroups(G, ll)]
 end
 
+"""
+    low_index_subgroups(G::Group, n::Int)
+
+Return an iterator of the subgroups of index at most `n` in `G`.
+Very likely it is better to use [`low_index_subgroup_classes`](@ref) instead.
+
+# Examples
+```jldoctest
+julia> G = alternating_group(6);
+
+julia> length(collect(low_index_subgroups(G, 6)))
+13
+```
+"""
+low_index_subgroups(G::T, n::Int) where T <: Union{GAPGroup, FinGenAbGroup} = Iterators.flatten(low_index_subgroup_classes(G, n))
+
 """
     conjugate_group(G::T, x::GAPGroupElem) where T <: GAPGroup
 
@@ -1254,6 +1304,22 @@ function hall_subgroup_classes(G::GAPGroup, P::AbstractVector{<:IntegerUnion})
    end
 end
 
+"""
+    hall_subgroups(G::Group, P::AbstractVector{<:IntegerUnion})
+
+Return an iterator of the Hall `P`-subgroups in `G`.
+Very likely it is better to use [`hall_subgroup_classes`](@ref) instead.
+
+# Examples
+```jldoctest
+julia> g = GL(3, 2);
+
+julia> describe(first(hall_subgroups(g, [2, 3])))
+"S4"
+```
+"""
+hall_subgroups(G::T, P::AbstractVector{<:IntegerUnion}) where T <: Union{GAPGroup, FinGenAbGroup} = Iterators.flatten(hall_subgroup_classes(G, P))
+
 @doc raw"""
     sylow_system(G::Group)
 
@@ -1304,6 +1370,22 @@ function complement_classes(G::T, N::T) where T <: GAPGroup
    end
 end
 
+@doc raw"""
+    complements(G::T, N::T) where T <: GAPGroup
+
+Return an iterator of the complements of the normal subgroup `N` in `G`.
+Very likely it is better to use [`complement_classes`](@ref) instead.
+
+# Examples
+```jldoctest
+julia> G = symmetric_group(3);
+
+julia> describe(first(complements(G, derived_subgroup(G)[1])))
+"C2"
+```
+"""
+complements(G::T, N::T) where T <: GAPGroup = Iterators.flatten(complement_classes(G, N))
+
 @doc raw"""
     complement_system(G::Group)
 

src/Groups/sub.jl

@@ -455,7 +455,7 @@ julia> lower_central_series(symmetric_group(4))
     upper_central_series(G::GAPGroup)
 
 Return the vector $[ G_1, G_2, \ldots ]$ where the last entry is the
-trivial group, and $G_i$ is defined as the overgroup of $G_{i+1}
+trivial group, and $G_i$ is defined as the overgroup of $G_{i+1}$
 satisfying $G_i / G_{i+1} = Z(G/G_{i+1})$. The series ends as soon as
 it is repeating (e.g. when the whole group $G$ is reached, which
 happens if and only if $G$ is nilpotent).
@@ -554,9 +554,12 @@ function is_maximal_subgroup(H::T, G::T; check::Bool = true) where T <: GAPGroup
   if check
     @req is_subset(H, G) "H is not a subgroup of G"
   end
-  if order(G) // order(H) < 100
+  ind = index(G, H)
+  is_prime(ind) && return true
+  if ind < 100
+    # Do not unpack the right transversal object.
     t = right_transversal(G, H)[2:end] #drop the identity
-    return all(x -> order(sub(G, vcat(gens(H), [x]))[1]) == order(G), t)
+    return all(i -> order(sub(G, vcat(gens(H), [t[i]]))[1]) == order(G), 2:ind)
   end
   return any(C -> H in C, maximal_subgroup_classes(G))
 end

src/exports.jl

@@ -357,6 +357,7 @@ export complement_of_point_ideal
 export complement_of_prime_ideal
 export complement_scheme
 export complement_system, has_complement_system, set_complement_system
+export complements
 export complete_bipartite_graph
 export complete_graph
 export complete_intersection_germ
@@ -611,6 +612,7 @@ export halfspace
 export halfspace_matrix_pair
 export hall_subgroup
 export hall_subgroup_classes
+export hall_subgroups
 export hall_system, has_hall_system, set_hall_system
 export has_du_val_singularities
 export has_edge
@@ -944,6 +946,7 @@ export localization
 export localized_ring
 export loops
 export low_index_subgroup_classes
+export low_index_subgroups
 export lower_central_series, has_lower_central_series, set_lower_central_series
 export lower_triangular_matrix
 export map
@@ -982,6 +985,7 @@ export maximal_groebner_cone
 export maximal_normal_subgroups, has_maximal_normal_subgroups, set_maximal_normal_subgroups
 export maximal_polyhedra, maximal_polyhedra_and_multiplicities
 export maximal_subgroup_classes, has_maximal_subgroup_classes, set_maximal_subgroup_classes
+export maximal_subgroups
 export metadata
 export milnor_algebra
 export milnor_number

test/Groups/GrpAb.jl

@@ -85,6 +85,7 @@ end
     # conjugacy classes of subgroups
     CC = subgroup_classes(G1)
     @test length(CC) == length(subgroup_classes(G2))
+    @test length(CC) == length(collect(subgroups(G1)))
     @test all(C -> length(C) == 1, CC)
     @test rand(CC[1]) == representative(CC[1])
     @test acting_group(CC[1]) == G1
@@ -117,10 +118,12 @@ end
       S1 = low_index_subgroup_classes(G1, n)
       S2 = low_index_subgroup_classes(G2, n)
       @test length(S1) == length(S2)
+      @test length(S1) == length(collect(low_index_subgroups(G1, n)))
     end
     S1 = maximal_subgroup_classes(G1)
     S2 = maximal_subgroup_classes(G2)
     @test sort!([length(x) for x in S1]) == sort!([length(x) for x in S2])
+    @test length(S1) == length(collect(maximal_subgroups(G1)))
 
     # operations
     x = representative(rand(cc))
@@ -140,6 +143,8 @@ end
     for P in subsets(Set(primes))
       @test [images(iso, representative(C))[1] for C in hall_subgroup_classes(G1, collect(P))] ==
             map(representative, hall_subgroup_classes(G2, collect(P)))
+      @test [images(iso, C)[1] for C in hall_subgroups(G1, collect(P))] ==
+            collect(hall_subgroups(G2, collect(P)))
     end
     @test sort!([order(images(iso, S)[1]) for S in hall_system(G1)]) ==
           sort!([order(S) for S in hall_system(G2)])

test/Groups/conjugation.jl

@@ -108,7 +108,9 @@
 
   G = symmetric_group(5)
   CC = @inferred maximal_subgroup_classes(G)
-  all(H -> degree(H) == degree(G), map(representative, CC))
+  @test all(H -> degree(H) == degree(G), map(representative, CC))
+  @test all(H -> is_maximal_subgroup(G, H), map(representative, CC))
+  @test !is_maximal_subgroup(G, trivial_subgroup(G)[1])
 
   G = symmetric_group(10)
   x = rand(G)