fourgroup

The four-group, also known as the Klein four group, is a small abelian group of order 4 that is not cyclic. It is typically denoted V4 or V and can be realized as the direct product Z2 × Z2, consisting of four elements that can be written as (0,0), (1,0), (0,1), and (1,1) with component-wise addition modulo 2. In this realization, every non-identity element has order 2, and the group operation is commutative.

As a structure, V4 is an elementary abelian 2-group of rank 2. It has exactly three subgroups of order 2, each generated by one of the three non-identity elements, and it contains the identity as its only element of order 1. The Cayley table reflects that the product of any two distinct non-identity elements is the third non-identity element, while each non-identity element squares to the identity.

The Klein four group appears in several standard contexts in abstract algebra and geometry. It is a normal subgroup of the symmetric group S4, consisting of the identity and the three double transpositions: {e, (12)(34), (13)(24), (14)(23)}. It is also the normal subgroup of the alternating group A4 with quotient A4/V4 ≅ C3. In geometry, D2, the dihedral group of order 4 (the symmetry group of a rectangle), is isomorphic to V4. It is also a classic example of a direct product of two copies of Z2.

The automorphism group Aut(V4) is isomorphic to S3, corresponding to the permutations of the three non-identity elements. The Klein four group serves as a fundamental example of a non-cyclic abelian group and of direct product structure in group theory.