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It happens that there are just two possible groups of 4 elements.  One is the cyclic group <math>Z_4\!</math> (from German ''Zyklus''), which this is not.  The other is Klein's four-group <math>V_4\!</math> (from German ''Vier''), which this is.

It happens that there are just two possible groups of 4 elements.  One is the cyclic group <math>Z_4\!</math> (from German ''Zyklus''), which this is not.  The other is the Klein four-group <math>V_4\!</math> (from German ''Vier''), which this is.

More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''.  One says that the orbits are preserved by the action of the group.  There is an ''Orbit Lemma'' of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group.  In this instance, <math>\operatorname{T}_{00}</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:  <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.</math>  Amazing!

More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''.  One says that the orbits are preserved by the action of the group.  There is an ''Orbit Lemma'' of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group.  In this instance, <math>\operatorname{T}_{00}</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:  <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.</math>  Amazing!