diff --git a/gtsam/base/concepts.h b/gtsam/base/concepts.h
index 967a91707..9011c759b 100644
--- a/gtsam/base/concepts.h
+++ b/gtsam/base/concepts.h
@@ -23,8 +23,9 @@ namespace traits {
  * @brief Associate a unique tag with each of the main GTSAM concepts
  */
 //@{
-template <class T>
-struct structure_category {}; // specializations should be derived from one of the following tags
+template<class T>
+struct structure_category;
+// specializations should be derived from one of the following tags
 //@}
 
 /**
@@ -32,147 +33,179 @@ struct structure_category {}; // specializations should be derived from one of t
  * @brief Possible values for traits::structure_category<T>::type
  */
 //@{
-struct manifold_tag {};
-struct group_tag {};
-struct lie_group_tag : public manifold_tag, public group_tag {};
-struct vector_space_tag : public lie_group_tag {};
+struct manifold_tag {
+};
+struct group_tag {
+};
+struct lie_group_tag: public manifold_tag, public group_tag {
+};
+struct vector_space_tag: public lie_group_tag {
+};
 //@}
 
-} // namespace traits
+}// namespace traits
 
 namespace traits {
 
 /** @name Manifold Traits */
 //@{
-template <class Manifold> struct TangentVector;
-template <class Manifold> struct DefaultChart;
+template<class Manifold> struct TangentVector;
+template<class Manifold> struct DefaultChart;
 //@}
 
-} // namespace traits
+}// namespace traits
 
 /*
-template<class T>
-class ManifoldConcept {
+ template<class T>
+ class ManifoldConcept {
  public:
-  typedef T Manifold;
-  typedef typename traits::TangentVector<T>::type TangentVector;
-  typedef typename traits::DefaultChart<T>::type DefaultChart;
-  static const size_t dim = traits::dimension<T>::value;
+ typedef T Manifold;
+ typedef typename traits::TangentVector<T>::type TangentVector;
+ typedef typename traits::DefaultChart<T>::type DefaultChart;
+ static const size_t dim = traits::dimension<T>::value;
 
-  BOOST_CONCEPT_USAGE(ManifoldConcept) {
-    BOOST_STATIC_ASSERT(boost::is_base_of<traits::manifold_tag, traits::structure<Manifold> >);
-    BOOST_STATIC_ASSERT(TangentVector::SizeAtCompileTime == dim);
-    // no direct usage for manifold since most usage is through a chart
-  }
+ BOOST_CONCEPT_USAGE(ManifoldConcept) {
+ BOOST_STATIC_ASSERT(boost::is_base_of<traits::manifold_tag, traits::structure<Manifold> >);
+ BOOST_STATIC_ASSERT(TangentVector::SizeAtCompileTime == dim);
+ // no direct usage for manifold since most usage is through a chart
+ }
  private:
-  Manifold p;
-  TangentVector v;
-};
+ Manifold p;
+ TangentVector v;
+ };
 
-template<class C>
-class ChartConcept {
+ template<class C>
+ class ChartConcept {
  public:
-  typedef C Chart;
-  typedef typename traits::Manifold<Chart>::type Manifold;
-  typedef typename traits::TangentVector<Manifold>::type TangentVector;
+ typedef C Chart;
+ typedef typename traits::Manifold<Chart>::type Manifold;
+ typedef typename traits::TangentVector<Manifold>::type TangentVector;
 
-  BOOST_CONCEPT_USAGE(ChartConcept) {
-    v = Chart::local(p,q); // returns local coordinates of q w.r.t. origin p
-    q = Chart::retract(p,v); // returns retracted update of p with v
-  }
+ BOOST_CONCEPT_USAGE(ChartConcept) {
+ v = Chart::local(p,q); // returns local coordinates of q w.r.t. origin p
+ q = Chart::retract(p,v); // returns retracted update of p with v
+ }
 
  private:
-  Manifold p,q;
-  TangentVector v;
+ Manifold p,q;
+ TangentVector v;
 
-};
-*/
+ };
+ */
+
+namespace group {
+
+/** @name Free functions any Group needs to define */
+//@{
+template<typename G> G compose(const G&g, const G& h);
+template<typename G> G between(const G&g, const G& h);
+template<typename G> G inverse(const G&g);
+//@}
 
 namespace traits {
 
 /** @name Group Traits */
 //@{
-template <class Group> struct identity {};
-template <class Group> struct group_flavor {};
+template<class G> struct identity;
+template<class G> struct flavor;
 //@}
 
 /** @name Group Flavor Tags */
 //@{
-struct additive_group_tag {};
-struct multiplicative_group_tag {};
+struct additive_tag {
+};
+struct multiplicative_tag {
+};
 //@}
 
-} // namespace traits
+} // \ namespace traits
+} // \ namespace group
 
-template<class G>
-class GroupConcept {
- public:
-  typedef G Group;
-  static const Group identity = traits::identity<G>::value;
+/**
+ * Group Concept
+ */
+template<typename G>
+class Group {
+public:
 
-  BOOST_CONCEPT_USAGE(GroupConcept) {
-    BOOST_STATIC_ASSERT(boost::is_base_of<traits::group_tag,typename traits::structure_category<Group>::type>::value );
-    Group ip = inverse(p);
-    Group pq = compose(p, q);
-    Group d = between(p, q);
-    test = equal(p, q);
-    test2 = operator_usage(p, q, traits::group_flavor<Group>::type);
+  typedef typename traits::structure_category<G>::type structure_category_tag;
+  typedef typename group::traits::identity<G>::value_type identity_value_type;
+  typedef typename group::traits::flavor<G>::type flavor_tag;
+
+  BOOST_CONCEPT_USAGE(Group) {
+    using group::compose;
+    using group::between;
+    using group::inverse;
+    BOOST_STATIC_ASSERT(
+        boost::is_base_of<traits::group_tag, structure_category_tag>::value);
+    e = group::traits::identity<G>::value;
+    d = compose(g, h);
+    d = between(g, h);
+    ig = inverse(g);
+    test = operator_usage(g, h, flavor);
   }
 
-  bool check_invariants(const Group& a, const Group& b) {
-    return (equal(compose(a, inverse(a)), identity))
-        && (equal(between(a, b), compose(inverse(a), b)))
-        && (equal(compose(a, between(a, b)), b))
-        && operator_usage(a, b, traits::group_flavor<Group>::type);
-  }
+// TODO: these all require default constructors :-(
+// Also, requires equal which is not required of a group
+//  Group():e(group::traits::identity<G>::value) {
+//  }
+//
+//  bool check_invariants(const G& a, const G& b) {
+//    return (equal(compose(a, inverse(a)), e))
+//        && (equal(between(a, b), compose(inverse(a), b)))
+//        && (equal(compose(a, between(a, b)), b))
+//        && operator_usage(a, b, flavor);
+//  }
 
- private:
-  Group p,q;
+private:
+  flavor_tag flavor;
+  G e, g, h, gh, ig, d;
   bool test, test2;
 
-  bool operator_usage(const Group& a, const Group& b, const traits::multiplicative_group_tag&) {
-    return equal(compose(a, b), a*b);
+  bool operator_usage(const G& a, const G& b,
+      group::traits::multiplicative_tag) {
+    return group::compose(a, b) == a * b;
 
   }
-  bool operator_usage(const Group& a, const Group& b, const traits::additive_group_tag&) {
-    return equal(compose(a, b), a+b);
+  bool operator_usage(const G& a, const G& b, group::traits::additive_tag) {
+    return group::compose(a, b) == a + b;
   }
 
 };
+
 /*
-template <class L>
-class LieGroupConcept : public GroupConcept<L>, public ManifoldConcept<L> {
+ template <class L>
+ class LieGroupConcept : public GroupConcept<L>, public ManifoldConcept<L> {
 
-  BOOST_CONCEPT_USAGE(LieGroupConcept) {
-    BOOST_STATIC_ASSERT(boost::is_base_of<traits::lie_group_tag, traits::structure<L> >);
-  }
-};
+ BOOST_CONCEPT_USAGE(LieGroupConcept) {
+ BOOST_STATIC_ASSERT(boost::is_base_of<traits::lie_group_tag, traits::structure<L> >);
+ }
+ };
 
-template <class V>
-class VectorSpaceConcept : public LieGroupConcept {
-  typedef typename traits::DefaultChart<V>::type Chart;
-  typedef typename GroupConcept<V>::identity identity;
+ template <class V>
+ class VectorSpaceConcept : public LieGroupConcept {
+ typedef typename traits::DefaultChart<V>::type Chart;
+ typedef typename GroupConcept<V>::identity identity;
 
-  BOOST_CONCEPT_USAGE(VectorSpaceConcept) {
-    BOOST_STATIC_ASSERT(boost::is_base_of<traits::vector_space_tag, traits::structure<L> >);
-    r = p+q;
-    r = -p;
-    r = p-q;
-  }
+ BOOST_CONCEPT_USAGE(VectorSpaceConcept) {
+ BOOST_STATIC_ASSERT(boost::is_base_of<traits::vector_space_tag, traits::structure<L> >);
+ r = p+q;
+ r = -p;
+ r = p-q;
+ }
 
-  bool check_invariants(const V& a, const V& b) {
-    return equal(compose(a, b), a+b)
-        && equal(inverse(a), -a)
-        && equal(between(a, b), b-a)
-        && equal(Chart::retract(a, b), a+b)
-        && equal(Chart::local(a, b), b-a);
-  }
+ bool check_invariants(const V& a, const V& b) {
+ return equal(compose(a, b), a+b)
+ && equal(inverse(a), -a)
+ && equal(between(a, b), b-a)
+ && equal(Chart::retract(a, b), a+b)
+ && equal(Chart::local(a, b), b-a);
+ }
 
  private:
-  V p,q,r;
-};
-*/
+ V g,q,r;
+ };
+ */
 
 } // namespace gtsam
 
-
diff --git a/gtsam/geometry/Cyclic.cpp b/gtsam/geometry/Cyclic.cpp
new file mode 100644
index 000000000..2b14ec6ba
--- /dev/null
+++ b/gtsam/geometry/Cyclic.cpp
@@ -0,0 +1,27 @@
+/* ----------------------------------------------------------------------------
+
+ * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
+ * Atlanta, Georgia 30332-0415
+ * All Rights Reserved
+ * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
+
+ * See LICENSE for the license information
+
+ * -------------------------------------------------------------------------- */
+
+/**
+ * @file   Cyclic.cpp
+ * @brief  Cyclic group implementation
+ * @author Frank Dellaert
+ **/
+
+#include <gtsam/geometry/Cyclic.h>
+
+namespace gtsam {
+namespace group {
+namespace traits {
+
+} // \namespace traits
+} // \namespace group
+} // \namespace gtsam
+
diff --git a/gtsam/geometry/Cyclic.h b/gtsam/geometry/Cyclic.h
new file mode 100644
index 000000000..85602d12d
--- /dev/null
+++ b/gtsam/geometry/Cyclic.h
@@ -0,0 +1,89 @@
+/* ----------------------------------------------------------------------------
+
+ * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
+ * Atlanta, Georgia 30332-0415
+ * All Rights Reserved
+ * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
+
+ * See LICENSE for the license information
+
+ * -------------------------------------------------------------------------- */
+
+/**
+ * @file   Cyclic.h
+ * @brief  Cyclic group, i.e., the integers modulo N
+ * @author Frank Dellaert
+ **/
+
+#include <gtsam/base/concepts.h>
+#include <cstddef>
+
+namespace gtsam {
+
+template<size_t N>
+class Cyclic {
+  size_t i_; ///< we just use an unsigned int
+public:
+  /// Constructor
+  Cyclic(size_t i) :
+      i_(i) {
+  }
+  /// Cast to size_t
+  operator size_t() const {
+    return i_;
+  }
+  /// Addition modulo N
+  Cyclic operator+(const Cyclic& h) const {
+    return (i_ + h.i_) % N;
+  }
+  /// Subtraction modulo N
+  Cyclic operator-(const Cyclic& h) const {
+    return (N + i_ - h.i_) % N;
+  }
+  /// Inverse
+  Cyclic operator-() const {
+    return (N - i_) % N;
+  }
+};
+
+namespace traits {
+/// Define Cyclic to be a model of the Group concept
+template<size_t N> struct structure_category<Cyclic<N> > {
+  typedef group_tag type;
+};
+} // \namespace gtsam::traits
+
+namespace group {
+
+template<size_t N>
+Cyclic<N> compose(const Cyclic<N>&g, const Cyclic<N>& h) {
+  return g + h;
+}
+
+template<size_t N>
+Cyclic<N> between(const Cyclic<N>&g, const Cyclic<N>& h) {
+  return h - g;
+}
+
+template<size_t N>
+Cyclic<N> inverse(const Cyclic<N>&g) {
+  return -g;
+}
+
+namespace traits {
+
+/// Define the trait that specifies Cyclic's identity element
+template<size_t N> struct identity<Cyclic<N> > {
+  static const Cyclic<N> value = Cyclic<N>(0);
+  typedef Cyclic<N> value_type;
+};
+
+/// Define the trait that asserts Cyclic is an additive group
+template<size_t N> struct flavor<Cyclic<N> > {
+  typedef additive_tag type;
+};
+
+} // \namespace gtsam::group::traits
+} // \namespace gtsam::group
+} // \namespace gtsam
+
diff --git a/gtsam/geometry/tests/testCyclic.cpp b/gtsam/geometry/tests/testCyclic.cpp
index f15654b2e..00ea2c853 100644
--- a/gtsam/geometry/tests/testCyclic.cpp
+++ b/gtsam/geometry/tests/testCyclic.cpp
@@ -15,53 +15,76 @@
  * @author Frank Dellaert
  **/
 
-#include <gtsam/base/concepts.h>
-#include <cstddef>
-
-namespace gtsam {
-
-template<size_t N>
-class Cyclic {
-  size_t i_;
-public:
-  static const Cyclic Identity = Cyclic(0);
-  Cyclic(size_t i) :
-      i_(i) {
-  }
-};
-
-namespace traits {
-template<size_t N> struct identity<Cyclic<N> > {
-  static const Cyclic<N> value = Cyclic<N>::Identity;
-  typedef Cyclic<N> value_type;
-};
-template<size_t N> struct group_flavor<Cyclic<N> > {
-  typedef additive_group_tag type;
-};
-} // \namespace traits
-
-} // \namespace gtsam
-
-//#include <gtsam/geometry/Cyclic.h>
+#include <gtsam/geometry/Cyclic.h>
 #include <gtsam/base/Testable.h>
 #include <CppUnitLite/TestHarness.h>
 
 using namespace std;
 using namespace gtsam;
 
-BOOST_CONCEPT_ASSERT((GroupConcept<Cyclic<6> >));
+typedef Cyclic<6> G; // Let's use the cyclic group of order 6
 
-/* ************************************************************************* */
+//******************************************************************************
+TEST(Cyclic, Concept) {
+  BOOST_CONCEPT_ASSERT((Group<G>));
+  EXPECT_LONGS_EQUAL(0, group::traits::identity<G>::value);
+  G g(2), h(3);
+  // EXPECT(Group<G>().check_invariants(g,h))
+}
+
+//******************************************************************************
 TEST(Cyclic, Constructor) {
-Cyclic<6> g(0);
-//  EXPECT(assert_equal(p1, p2));
-//  EXPECT_LONGS_EQUAL(2,offset2.size());
+  G g(0);
 }
 
-/* ************************************************************************* */
+//******************************************************************************
+TEST(Cyclic, Compose) {
+  EXPECT_LONGS_EQUAL(0, group::compose(G(0),G(0)));
+  EXPECT_LONGS_EQUAL(1, group::compose(G(0),G(1)));
+  EXPECT_LONGS_EQUAL(2, group::compose(G(0),G(2)));
+  EXPECT_LONGS_EQUAL(3, group::compose(G(0),G(3)));
+  EXPECT_LONGS_EQUAL(4, group::compose(G(0),G(4)));
+  EXPECT_LONGS_EQUAL(5, group::compose(G(0),G(5)));
+
+  EXPECT_LONGS_EQUAL(2, group::compose(G(2),G(0)));
+  EXPECT_LONGS_EQUAL(3, group::compose(G(2),G(1)));
+  EXPECT_LONGS_EQUAL(4, group::compose(G(2),G(2)));
+  EXPECT_LONGS_EQUAL(5, group::compose(G(2),G(3)));
+  EXPECT_LONGS_EQUAL(0, group::compose(G(2),G(4)));
+  EXPECT_LONGS_EQUAL(1, group::compose(G(2),G(5)));
+}
+
+//******************************************************************************
+TEST(Cyclic, Between) {
+  EXPECT_LONGS_EQUAL(0, group::between(G(0),G(0)));
+  EXPECT_LONGS_EQUAL(1, group::between(G(0),G(1)));
+  EXPECT_LONGS_EQUAL(2, group::between(G(0),G(2)));
+  EXPECT_LONGS_EQUAL(3, group::between(G(0),G(3)));
+  EXPECT_LONGS_EQUAL(4, group::between(G(0),G(4)));
+  EXPECT_LONGS_EQUAL(5, group::between(G(0),G(5)));
+
+  EXPECT_LONGS_EQUAL(4, group::between(G(2),G(0)));
+  EXPECT_LONGS_EQUAL(5, group::between(G(2),G(1)));
+  EXPECT_LONGS_EQUAL(0, group::between(G(2),G(2)));
+  EXPECT_LONGS_EQUAL(1, group::between(G(2),G(3)));
+  EXPECT_LONGS_EQUAL(2, group::between(G(2),G(4)));
+  EXPECT_LONGS_EQUAL(3, group::between(G(2),G(5)));
+}
+
+//******************************************************************************
+TEST(Cyclic, Ivnverse) {
+  EXPECT_LONGS_EQUAL(0, group::inverse(G(0)));
+  EXPECT_LONGS_EQUAL(5, group::inverse(G(1)));
+  EXPECT_LONGS_EQUAL(4, group::inverse(G(2)));
+  EXPECT_LONGS_EQUAL(3, group::inverse(G(3)));
+  EXPECT_LONGS_EQUAL(2, group::inverse(G(4)));
+  EXPECT_LONGS_EQUAL(1, group::inverse(G(5)));
+}
+
+//******************************************************************************
 int main() {
-TestResult tr;
-return TestRegistry::runAllTests(tr);
+  TestResult tr;
+  return TestRegistry::runAllTests(tr);
 }
-/* ************************************************************************* */
+//******************************************************************************