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  "body": "\n## Description\n\nThe circle group is the set $\\mathbb{T} = \\{z \\in \\mathbb{C} : |z| = 1\\}$ of the complex numbers of absolute value $1$, with complex multiplication.\nIt is an [[abelian group|082GHNPJ53]] with identity element $1$, and the inverse of $z$ is its complex conjugate $\\bar{z} = 1/z$.\nEach element of $\\mathbb{T}$ is $e^{i\\theta}$ for a real number $\\theta$, and $e^{i\\theta} e^{i\\varphi} = e^{i(\\theta + \\varphi)}$.\n\n## Remarks\n\nThe map $t \\mapsto e^{2\\pi i t}$ is a surjective homomorphism from $(\\mathbb{R}, +)$ to $\\mathbb{T}$ with kernel $\\mathbb{Z}$, thus $\\mathbb{T}$ is isomorphic to the quotient $\\mathbb{R}/\\mathbb{Z}$.\nThe circle group is also isomorphic to the group $\\mathrm{SO}(2)$ of rotations of the plane about the origin.\nFor each positive integer $n$, the [[group of $n$th roots of unity|0WX11EA3F3]] is a finite subgroup of $\\mathbb{T}$.\n\n## Explanations\n\n### L1 human\n\nThe circle group is the unit circle in the complex plane, where the product of two points is their complex product.\nIf you write the points as angles, the product adds the angles.\nThus the circle group is the group of rotations of a circle.\n",
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