diff --git a/Project.toml b/Project.toml
index 513fe43..782e45f 100644
--- a/Project.toml
+++ b/Project.toml
@@ -5,6 +5,7 @@ version = "0.5.2"
 
 [deps]
 AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
+DataStructures = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
 GroupsCore = "d5909c97-4eac-4ecc-a3dc-fdd0858a4120"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 ThreadsX = "ac1d9e8a-700a-412c-b207-f0111f4b6c0d"
diff --git a/src/Groups.jl b/src/Groups.jl
index fcfdd3e..e91e8eb 100644
--- a/src/Groups.jl
+++ b/src/Groups.jl
@@ -22,6 +22,8 @@ include("arithmetic.jl")
 include("findreplace.jl")
 
 module New
+using DataStructures
+
 include("new_types.jl")
 include("new_hashing.jl")
 include("normalform.jl")
diff --git a/src/iteration.jl b/src/iteration.jl
new file mode 100644
index 0000000..89ce140
--- /dev/null
+++ b/src/iteration.jl
@@ -0,0 +1,55 @@
+mutable struct FPIterState{GEl, T}
+    elts::OrderedSet{GEl}
+    u::GEl
+    v::GEl
+    elts_iter_state::T
+end
+
+Base.in(x, itr::FPIterState) = x in itr.elts
+Base.push!(itr::FPIterState, x) = push!(itr.elts, x)
+function FPIterState(G::AbstractFPGroup)
+    S = OrderedSet([one(G)])
+    elt, state = iterate(S)
+    return elt, FPIterState(S, one(G), one(G), state)
+end
+
+function Base.iterate(G::AbstractFPGroup)
+    elt, state = FPIterState(G)
+    return one(G), (state, elt, 1)
+end
+
+function Base.iterate(G::AbstractFPGroup, state)
+    iter, elt, gen_idx = state
+
+    if gen_idx > length(alphabet(G))
+        res = iterate(iter.elts, iter.elts_iter_state)
+        res === nothing && return nothing
+        gen_idx = 1
+        elt = first(res)
+        iter.elts_iter_state = last(res)
+    end
+
+    res = let (u, v) = (iter.u, iter.v), elt = elt
+        copyto!(v, elt) # this invalidates normalform of v
+        push!(word(v), gen_idx)
+        resize!(word(u), 0)
+
+        _setnormalform!(v, false)
+        _setvalidhash!(v, false)
+
+        @assert !isnormalform(v)
+        normalform!(u, v)
+    end
+
+    if res in iter
+        return iterate(G, (iter, elt, gen_idx+1))
+    else
+        w = deepcopy(res)
+        @assert isnormalform(w)
+        push!(iter, w)
+        return w, (iter, elt, gen_idx+1)
+    end
+end
+
+# Groups.Core default:
+# Base.IteratorSize(::Type{<:AbstractFPGroup}) = Base.SizeUnknown()
diff --git a/src/new_types.jl b/src/new_types.jl
index d05b030..8241414 100644
--- a/src/new_types.jl
+++ b/src/new_types.jl
@@ -43,50 +43,7 @@ Base.one(G::AbstractFPGroup) = FPGroupElement(one(word_type(G)), G)
 Base.eltype(::Type{FPG}) where {FPG<:AbstractFPGroup} =
     FPGroupElement{FPG, word_type(FPG)}
 
-struct FPGroupIter{GEl}
-    elts::Vector{GEl}
-    seen::Set{GEl}
-    u::GEl
-    v::GEl
-end
-
-FPGroupIter(G::AbstractFPGroup) =
-    FPGroupIter([one(G)], Set([one(G)]), one(G), one(G))
-
-Base.iterate(G::AbstractFPGroup) = one(G), (FPGroupIter(G), 1, 1)
-@inline function Base.iterate(G::AbstractFPGroup, state)
-    iter, elt_idx, gen_idx = state
-
-    if gen_idx > length(alphabet(G))
-        elt_idx == length(iter.elts) && return nothing
-        gen_idx = 1
-        elt_idx += 1
-    end
-
-
-    res = let (u, v) = (iter.u, iter.v), elt = iter.elts[elt_idx]
-        copyto!(v, elt) # this invalidates normalform of v
-        @assert !isnormalform(v)
-        push!(word(v), gen_idx)
-        resize!(word(u), 0)
-
-        normalform!(u, v)
-    end
-
-    if res in iter.seen
-        return iterate(G, (iter, elt_idx, gen_idx+1))
-    else
-        w = deepcopy(res)
-        @assert isnormalform(w)
-        push!(iter.elts, w)
-        push!(iter.seen, w)
-        state = (iter, elt_idx, gen_idx+1)
-        return w, state
-    end
-end
-
-# the default:
-# Base.IteratorSize(::Type{<:AbstractFPGroup}) = Base.SizeUnknown()
+include("iteration.jl")
 
 GroupsCore.ngens(G::AbstractFPGroup) = length(G.gens)