Z2

Z2 is a notation encountered in mathematics and related disciplines that can refer to two related two-element algebraic structures, depending on context. In many texts it denotes either the cyclic group of order 2 under addition or the two-element field used in modular arithmetic, commonly written as Z/2Z or GF(2).

As a group, Z2 consists of two elements, 0 and 1, with addition performed modulo 2: 0+0=0, 0+1=1, and 1+1=0. It is the simplest nontrivial cyclic group and is isomorphic to the additive group of Z/2Z. This structure is fundamental in parity considerations, binary arithmetic, and various symmetry arguments.

As a field, Z2 is the same two-element set equipped with both addition and multiplication modulo 2, making GF(2). It has characteristic 2, and its nonzero element (1) is its own multiplicative inverse. GF(2) is the smallest finite field and serves as the base field for many applications, including linear algebra over finite fields, error-correcting codes, cryptography, and digital logic, where computations are performed modulo 2.

Notationally, the subscript 2 in Z2 reflects modulo 2 or order 2, and some authors use Z2 to denote the ring Z/2Z, while others reserve Z2 for the additive group. The precise meaning is determined by the mathematical context.