Z2Gs

Z2Gs is a term used in mathematical literature to denote a class of structures that pair a base group G with a Z2-grading, where Z2 is the cyclic group of order two. The plural form indicates a family of such objects, often arising when a group carries a natural parity decomposition. Because terminology varies by field, the exact definition of Z2G can differ, but a common thread is the incorporation of a binary symmetry or grading together with a group structure.

A typical definition treats a Z2G as a group G equipped with a surjective homomorphism ε: G → Z2. The kernel H = ker ε is a normal subgroup of index 2, yielding a decomposition G0 = H and G1 = ε−1(1) into even and odd parts. The grading respects multiplication: G0G0 ⊆ G0, G0G1 ⊆ G1, G1G0 ⊆ G1, and G1G1 ⊆ G0. Equivalently, G is a Z2-graded group determined by the homomorphism ε, with quotient G/H ≅ Z2.

Examples include any group with an index-2 normal subgroup, such as the dihedral group D4, where the rotations form the even part and reflections form the odd part. More generally, any G with a projection to G/H ≅ Z2 yields a Z2G structure.

Applications of Z2G concepts appear in representation theory of graded groups, topology and group cohomology, and physics for parity symmetries and Z2 gauge structures. In other contexts, Z2G may refer to Z2-graded algebras or Z2-gauge theories, depending on convention and field.

See also: Z2-graded algebra, semidirect product, parity, dihedral group, group cohomology.