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  text: 'A **subgroup** of a [-3gd] $(G,*)$ is a group of the form $(H,*)$, where $H \\subset G$. We usually say simply that $H$ is a subgroup of $G$. \n\nFor a subset of a group $G$ to be a subgroup, it needs to satisfy all of the group axioms itself: [3gy closure], [3h4 associativity], [54p identity], and [inverse_element inverse]. We get associativity for free because $G$ is a group. So the requirements of a subgroup $H$ are:\n\n1. **[3gy Closure]:** For any $x, y$ in $H$, $x*y$ is in $H$.\n2. **[54p Identity]:** The identity $e$ of $G$ is in $H$.\n3.  **[inverse_element Inverses]:** For any $x$ in $H$, $x^{-1}$ is also in $H$.\n\nA subgroup is called [4h6 normal] if it is closed under [4gk conjugation]. \n\nThe subgroup [-3nt] is [573 transitive]: if $H$ is a subgroup of $G$, and $I$ is a subgroup of $H$, then $I$ is a subgroup of $G$.\n\n#Examples\n\nAny group is a subgroup of itself. The [-trivial_group] is a subgroup of every group. \n\nFor any [-48l] $n$, the set of multiples of $n$ is a subgroup of the integers (under [-addition]).',
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