Z2×Z2

The group Z₂ × Z₂ (read as "Z two direct product Z two") is the direct product of two copies of the cyclic group of order 2, denoted Z₂. It is a fundamental example of an abelian group with four elements, often studied in abstract algebra and group theory.

In Z₂, the elements are {0, 1} with addition modulo 2, where 1 + 1 = 0. The direct product Z₂ × Z₂ consists of ordered pairs (a, b), where a and b ∈ {0, 1}. The group operation is component-wise addition modulo 2, meaning (a, b) + (c, d) = (a + c mod 2, b + d mod 2). This results in four distinct elements: (0, 0), (0, 1), (1, 0), and (1, 1).

As an abelian group, Z₂ × Z₂ is isomorphic to the Klein four-group, a non-cyclic group where every non-identity element has order 2. It serves as a simple model for understanding properties like commutativity, subgroup structure, and quotient groups. The group is also isomorphic to the additive group of the field F₂², the 2-dimensional vector space over the finite field with two elements.

In terms of presentation, Z₂ × Z₂ can be described by the generators x and y with relations x² = 1, y² = 1, and xy = yx. This reflects its commutative nature and the fact that every element is its own inverse. The group is also a classic example of a non-cyclic group that is not isomorphic to Z₄, the cyclic group of order 4.

Applications of Z₂ × Z₂ appear in coding theory, cryptography, and combinatorial designs, where its symmetry and structure provide useful frameworks for analysis. Its simplicity makes it a key building block in the study of more complex algebraic systems.