In physics and mathematics, the Pauli group G 1 {\displaystyle G_{1}} on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I {\displaystyle I} and all of the Pauli matrices X = σ 1 =, Y = σ 2 =, Z = σ 3 = {\displaystyle X=\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad Y=\sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad Z=\sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}},together with the products of these matrices with the factors ± 1 {\displaystyle \pm 1} and ± i {\displaystyle \pm i} : G 1 = d e f { ± I, ± i I, ± X, ± i X, ± Y, ± i Y, ± Z, ± i Z } ≡ ⟨ X, Y, Z ⟩ {\displaystyle G_{1}\ {\stackrel {\mathrm {def} }{=}}\ \{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\}\equiv \langle X,Y,Z\rangle }. Wikipedia