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text: 'Given a subgroup $H$ of [-3gd] $G$, the *left cosets* of $H$ in $G$ are sets of the form $\\{ gh : h \\in H \\}$, for some $g \\in G$.\nThis is written $gH$ as a shorthand.\n\nSimilarly, the *right cosets* are the sets of the form $Hg = \\{ hg: h \\in H \\}$.\n\n# Examples\n%%%knows-requisite([497]):\n## Symmetric group\n\nIn $S_3$, the [-497] on three elements, we can list the elements as $\\{ e, (123), (132), (12), (13), (23) \\}$, using [49f cycle notation].\nDefine $A_3$ (which happens to have a name: the [-4hf]) to be the subgroup with elements $\\{ e, (123), (132) \\}$.\n\nThen the coset $(12) A_3$ has elements $\\{ (12), (12)(123), (12)(132) \\}$, which is simplified to $\\{ (12), (23), (13) \\}$.\n\nThe coset $(123)A_3$ is simply $A_3$, because $A_3$ is a subgroup so is closed under the group operation. $(123)$ is already in $A_3$.\n%%%\n\n[todo: more examples, with different requirements]\n\n# Properties\n\n- The left cosets of $H$ in $G$ [set_partition partition] $G$. ([4j5 Proof.])\n- For any pair of left cosets of $H$, there is a [499 bijection] between them; that is, all the cosets are all the same size. ([4j8 Proof.])\n\n# Why are we interested in cosets?\n\nUnder certain conditions (namely that the subgroup $H$ must be [4h6 normal]), we may define the [-quotient_group], a very important concept; see the page on [4j5 "left cosets partition the parent group"] for a glance at why this is useful.\n[todo: there must be a less clumsy way to do it]\n\nAdditionally, there is a key theorem whose usual proof considers cosets ([lagrange_theorem_on_subgroup_size Lagrange's theorem]) which strongly restricts the possible sizes of subgroups of $G$, and which itself is enough to classify all the groups of [3gg order] $p$ for $p$ [prime_number prime].\nLagrange's theorem also has very common applications in [-number_theory], in the form of the [fermat_euler_theorem Fermat-Euler theorem].',
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