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  "body": "\n## Description\n\nThe modular group is the [[special linear group|0V9PMN0RBH]]\n$$\\mathrm{SL}_2(\\Z) = \\set{\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix} : a, b, c, d \\in \\Z, \\ ad - bc = 1}$$\nunder matrix multiplication.\nIt acts on the [[upper half-plane|X6XGGH5FPW]] $\\mathbb{H}$ by [[Möbius transformations|0SG74WBFA2]] $\\tau \\mapsto (a\\tau + b)/(c\\tau + d)$.\nThe matrices $I$ and $-I$ both act as the identity map of $\\mathbb{H}$, so many authors call the [[quotient group|0KYX2N69J0]] $\\mathrm{PSL}_2(\\Z) = \\mathrm{SL}_2(\\Z) / \\set{\\pm I}$ the modular group.\nThe group is generated by\n$$S = \\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}, \\qquad T = \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix},$$\nwhich act by $S\\tau = -1/\\tau$ and $T\\tau = \\tau + 1$.\n\n## Remarks\n\nThe generators satisfy $S^2 = -I$ and $(ST)^3 = -I$.\nCheck: $ST = \\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 1 \\end{pmatrix}$, $(ST)^2 = \\begin{pmatrix} -1 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$, and $(ST)^3 = \\begin{pmatrix} -1 \u0026 0 \\\\ 0 \u0026 -1 \\end{pmatrix}$.\nThe group $\\mathrm{PSL}_2(\\Z)$ is the free product of a cyclic group of order $2$ and a cyclic group of order $3$.\nThe set $\\set{\\tau \\in \\mathbb{H} : \\abs{\\tau} \\ge 1, \\ \\abs{\\operatorname{Re} \\tau} \\le 1/2}$ is a fundamental domain for its action on $\\mathbb{H}$.\nThe [[modular forms|GWKN5VG78A]] are the functions on $\\mathbb{H}$ with a prescribed behaviour under this group.\n\n## Explanations\n\n### L1 human\n\nThe modular group is the group of $2 \\times 2$ integer matrices with determinant $1$.\nEach such matrix moves the points of the upper half-plane by a fraction $(a\\tau + b)/(c\\tau + d)$.\nEvery such move is built from two simple ones: the shift $\\tau \\mapsto \\tau + 1$ and the inversion $\\tau \\mapsto -1/\\tau$.\nThe group also describes the ways to change an oriented basis of a lattice in the plane.\n",
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