diff --git a/src/PropertyT.jl b/src/PropertyT.jl
index a60015f..ba9dc02 100644
--- a/src/PropertyT.jl
+++ b/src/PropertyT.jl
@@ -4,6 +4,7 @@ using JLD
 using GroupRings
 using Memento
 
+using Groups
 import Nemo: Group, GroupElem
 
 const logger = Memento.config("info", fmt="{msg}")
@@ -45,23 +46,21 @@ function ΔandSDPconstraints(name::String, G::Group)
     return Δ, sdp_constraints
 end
 
-function ΔandSDPconstraints{T<:GroupElem}(name::String, S::Vector{T}, radius::Int)
-   S, Id = generating_set()
+function ΔandSDPconstraints{T<:GroupElem}(name::String, S::Vector{T}, Id::T; radius::Int=2)
    info(logger, "Computing pm, Δ, sdp_constraints...")
-   t = @timed Δ, sdp_constraints = ΔandSDPconstraints(S, radius)
-   info(logger, timed_msg(t))
+   Δ, sdp_constraints = ΔandSDPconstraints(S, Id, radius=radius)
    pm_fname, Δ_fname = pmΔfilenames(name)
    save(pm_fname, "pm", parent(Δ).pm)
    save(Δ_fname, "Δ", Δ.coeffs)
+   return Δ, sdp_constraints
 end
 
-function ΔandSDPconstraints{T<:GroupElem}(S::Vector{T}, r::Int=2)
-    Id = parent(S[1])()
-    B, sizes = Groups.generate_balls(S, Id, radius=2*r)
+function ΔandSDPconstraints{T<:GroupElem}(S::Vector{T}, Id::T; radius::Int=2)
+    B, sizes = Groups.generate_balls(S, Id, radius=2*radius)
     info(logger, "Generated balls of sizes $sizes")
 
     info(logger, "Creating product matrix...")
-    t = @timed pm = GroupRings.create_pm(B, GroupRings.reverse_dict(B), sizes[r]; twisted=true)
+    t = @timed pm = GroupRings.create_pm(B, GroupRings.reverse_dict(B), sizes[radius]; twisted=true)
     info(logger, timed_msg(t))
 
     info(logger, "Creating sdp_constratints...")
@@ -70,7 +69,7 @@ function ΔandSDPconstraints{T<:GroupElem}(S::Vector{T}, r::Int=2)
 
     RG = GroupRing(parent(Id), B, pm)
 
-    Δ = splaplacian(RG, S, B[1:sizes[r]], sizes[2*r])
+    Δ = splaplacian(RG, S, Id, sizes[2*radius])
     return Δ, sdp_constraints
 end
 
@@ -143,6 +142,7 @@ end
 Kazhdan_from_sgap(λ,N) = sqrt(2*λ/N)
 
 function setup_logging(name::String)
+   isdir(name) || mkdir(name)
 
    Memento.add_handler(logger,
       Memento.DefaultHandler(joinpath(name,"full_$(string((now()))).log"),
@@ -155,20 +155,16 @@ function setup_logging(name::String)
 end
 
 
-function check_property_T(name::String, S, solver, upper_bound, tol, radius)
+function check_property_T(name::String, S, Id, solver, upper_bound, tol, radius)
 
-    if !isdir(name)
-        mkdir(name)
-    end
-
-    setup_logging(name)
+    isdir(name) || mkdir(name)
 
     if all(isfile.(pmΔfilenames(name)))
         # cached
         Δ, sdp_constraints = ΔandSDPconstraints(name, parent(S[1]))
     else
         # compute
-        Δ, sdp_constraints = ΔandSDPconstraints(name, S, radius)
+        Δ, sdp_constraints = ΔandSDPconstraints(name, S, Id, radius=radius)
     end
 
     info(logger, "|S| = $(length(S))")
@@ -202,7 +198,7 @@ function check_property_T(name::String, S, solver, upper_bound, tol, radius)
       end
       if sgap > 0
            info(logger, "λ ≥ $(Float64(trunc(sgap,12)))")
-            Kazhdan_κ = Kazhdan_from_sgap(sgap, S)
+            Kazhdan_κ = Kazhdan_from_sgap(sgap, length(S))
             Kazhdan_κ = Float64(trunc(Kazhdan_κ, 12))
             info(logger, "κ($name, S) ≥ $Kazhdan_κ: Group HAS property (T)!")
             return true
diff --git a/src/checksolution.jl b/src/checksolution.jl
index 4e04c92..2db075e 100644
--- a/src/checksolution.jl
+++ b/src/checksolution.jl
@@ -81,7 +81,7 @@ function distance_to_cone{T<:Rational, S<:Interval}(λ::T, sqrt_matrix::Array{S,
     SOS = compute_SOS(sqrt_matrix, Δ)
     info(logger, "ɛ(∑ξᵢ*ξᵢ) ∈ $(GroupRings.augmentation(SOS))")
     λ_int = @interval(λ)
-    Δ_int = GroupRingElem([@interval(c) for c in Δ.coeffs], parent(Δ).pm)
+    Δ_int = GroupRingElem([@interval(c) for c in Δ.coeffs], parent(Δ))
     SOS_diff = EOI(Δ_int, λ_int) - SOS
     eoi_SOS_L1_dist = norm(SOS_diff,1)
 
@@ -91,7 +91,7 @@ function distance_to_cone{T<:Rational, S<:Interval}(λ::T, sqrt_matrix::Array{S,
     info(logger, "ɛ(Δ² - λΔ - ∑ξᵢ*ξᵢ) ∈ $(ɛ_dist)")
     info(logger, "‖Δ² - λΔ - ∑ξᵢ*ξᵢ‖₁ ∈ $(eoi_SOS_L1_dist)")
 
-    distance_to_cone = λ - 2^(len-1)*eoi_SOS_L₁_dist
+    distance_to_cone = λ - 2^(len-1)*eoi_SOS_L1_dist
     return distance_to_cone
 end
 
@@ -99,14 +99,14 @@ function distance_to_cone{T<:AbstractFloat}(λ::T, sqrt_matrix::Array{T,2}, Δ::
     SOS = compute_SOS(sqrt_matrix, Δ)
 
     SOS_diff = EOI(Δ, λ) - SOS
-    eoi_SOS_L₁_dist = norm(SOS_diff,1)
+    eoi_SOS_L1_dist = norm(SOS_diff,1)
 
     info(logger, "λ = $λ")
     ɛ_dist = GroupRings.augmentation(SOS_diff)
     info(logger, "ɛ(Δ² - λΔ - ∑ξᵢ*ξᵢ) ≈ $(@sprintf("%.10f", ɛ_dist))")
-    info(logger, "‖Δ² - λΔ - ∑ξᵢ*ξᵢ‖₁ ≈ $(@sprintf("%.10f", eoi_SOS_L₁_dist))")
+    info(logger, "‖Δ² - λΔ - ∑ξᵢ*ξᵢ‖₁ ≈ $(@sprintf("%.10f", eoi_SOS_L1_dist))")
 
-    distance_to_cone = λ - 2^(len-1)*eoi_SOS_L₁_dist
+    distance_to_cone = λ - 2^(len-1)*eoi_SOS_L1_dist
     return distance_to_cone
 end
 
@@ -115,7 +115,7 @@ function check_distance_to_positive_cone(Δ::GroupRingElem, λ, P;
 
     isapprox(eigvals(P), abs(eigvals(P)), atol=tol) ||
         warn("The solution matrix doesn't seem to be positive definite!")
-    @assert P == Symmetric(P)
+   #  @assert P == Symmetric(P)
     Q = real(sqrtm(P))
 
     info(logger, "------------------------------------------------------------")
@@ -130,6 +130,7 @@ function check_distance_to_positive_cone(Δ::GroupRingElem, λ, P;
         return fp_distance
     end
 
+    info(logger, "")
     info(logger, "Projecting columns of rationalized Q to the augmentation ideal...")
     δ = eps(λ)
     Q_ℚ = ℚ(Q, δ)
@@ -140,20 +141,20 @@ function check_distance_to_positive_cone(Δ::GroupRingElem, λ, P;
 
     info(logger, "Checking in interval arithmetic")
     Q_ℚω_int = Float64.(Q_ℚω) ± δ
-    t = @timed Interval_dist_to_Σ² = distance_to_cone(λ_ℚ, Q_ℚω_int, Δ_ℚ, len=len)
+    t = @timed Interval_dist_to_ΣSq = distance_to_cone(λ_ℚ, Q_ℚω_int, Δ_ℚ, len=len)
     info(logger, timed_msg(t))
-    info(logger, "The Augmentation-projected actual distance (to positive cone) ∈ $(Interval_dist_to_Σ²)")
+    info(logger, "The Augmentation-projected actual distance (to positive cone) ∈ $(Interval_dist_to_ΣSq)")
     info(logger, "------------------------------------------------------------")
 
-    if Interval_dist_to_Σ².lo ≤ 0 || !rational
-        return Interval_dist_to_Σ²
+    if Interval_dist_to_ΣSq.lo ≤ 0 || !rational
+        return Interval_dist_to_ΣSq
     else
         info(logger, "Checking Projected SOS decomposition in exact rational arithmetic...")
-        t = @timed ℚ_dist_to_Σ² = distance_to_cone(λ_ℚ, Q_ℚω, Δ_ℚ, len=len)
+        t = @timed ℚ_dist_to_ΣSq = distance_to_cone(λ_ℚ, Q_ℚω, Δ_ℚ, len=len)
         info(logger, timed_msg(t))
-        @assert isa(ℚ_dist_to_Σ², Rational)
-        info(logger, "Augmentation-projected rational distance (to positive cone) ≥ $(Float64(trunc(ℚ_dist_to_Σ²,8)))")
+        @assert isa(ℚ_dist_to_ΣSq, Rational)
+        info(logger, "Augmentation-projected rational distance (to positive cone) ≥ $(Float64(trunc(ℚ_dist_to_ΣSq,8)))")
         info(logger, "------------------------------------------------------------")
-        return ℚ_dist_to_Σ²
+        return ℚ_dist_to_ΣSq
     end
 end
diff --git a/src/sdps.jl b/src/sdps.jl
index 365b960..20489a1 100644
--- a/src/sdps.jl
+++ b/src/sdps.jl
@@ -13,9 +13,9 @@ function constraints_from_pm(pm, total_length=maximum(pm))
     return constraints
 end
 
-function splaplacian(RG::GroupRing, S, basis, n=length(basis), T::Type=Int)
+function splaplacian(RG::GroupRing, S, Id=RG.group(), n=length(basis),T::Type=Int)
     result = RG(spzeros(T, n))
-    result[RG.group()] = T(length(S))
+    result[Id] = T(length(S))
     for s in S
         result[s] -= one(T)
     end
@@ -27,7 +27,7 @@ function create_SDP_problem(Δ::GroupRingElem, matrix_constraints; upper_bound=I
     Δ² = Δ*Δ
     @assert length(Δ.coeffs) == length(matrix_constraints)
     m = JuMP.Model();
-    JuMP.@variable(m, P[1:N, 1:N], SDP)
+    JuMP.@variable(m, P[1:N, 1:N])
     JuMP.@SDconstraint(m, P >= 0)
     JuMP.@constraint(m, sum(P[i] for i in eachindex(P)) == 0)