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  "body": "\n## Definition\n\nLet $\\Omega = \\set{1, \\dots, n}$ and let $G \\le \\operatorname{Sym}(\\Omega)$ be the [[permutation group|ZDJQRKPWGA]] [[generated|0RP7VWQ0YN]] by a finite set $S$ of permutations.\nA *base* of $G$ is a sequence $(b_1, \\dots, b_k)$ of points of $\\Omega$ such that only the identity of $G$ fixes every $b_i$.\nIt gives the *stabiliser chain* $G = G^{(1)} \\ge G^{(2)} \\ge \\dots \\ge G^{(k+1)} = \\set{e}$, where $G^{(i+1)}$ is the pointwise [[stabiliser|0AYMP41THN]] of $b_1, \\dots, b_i$ in $G$.\nA *strong generating set* for this base is a subset $T \\subseteq G$ such that $T \\cap G^{(i)}$ generates $G^{(i)}$ for each $i = 1, \\dots, k$.\n\nThe *Schreier–Sims algorithm* computes, from $S$, a base of $G$ and a strong generating set for it.\nIt builds the chain from the top, and it repeats these steps until no new generator appears:\n\n1. For each level $i$, compute the [[orbit|0YQPZB5Q1G]] $\\Delta_i = G^{(i)} b_i$ and, for each $\\delta \\in \\Delta_i$, an element $u_\\delta \\in G^{(i)}$ with $u_\\delta b_i = \\delta$.\n2. By Schreier's lemma, $G^{(i+1)}$ is generated by the *Schreier generators* $u_{s\\delta}^{-1} \\, s \\, u_\\delta$, with $s$ in the current generators of $G^{(i)}$ and $\\delta \\in \\Delta_i$.\n3. Sift each Schreier generator through the levels below $i$, that is, divide it by the coset representatives $u_\\delta$ found there. If the remainder is not the identity, add it to the generators of the level where sifting stopped, with a new base point if needed.\n\n## Remarks\n\nWhen the algorithm ends, $\\abs{G} = \\abs{\\Delta_1} \\abs{\\Delta_2} \\cdots \\abs{\\Delta_k}$, because $\\abs{\\Delta_i}$ is the [[index|0DTXBQPA2S]] $[G^{(i)} : G^{(i+1)}]$ by the orbit-stabiliser theorem.\nA permutation $\\sigma$ lies in $G$ exactly when sifting $\\sigma$ through the whole chain ends at the identity.\n\nExample: $G = S_3$ generated by $S = \\set{(1\\,2\\,3), (1\\,2)}$, with base $(1, 2)$.\nThe orbit of $1$ is $\\Delta_1 = \\set{1, 2, 3}$.\nThe stabiliser of $1$ is $G^{(2)} = \\set{e, (2\\,3)}$, and the orbit of $2$ under it is $\\Delta_2 = \\set{2, 3}$.\nSo $\\set{(1\\,2\\,3), (1\\,2), (2\\,3)}$ is a strong generating set, and $\\abs{G} = 3 \\cdot 2 = 6$.\n\nNon-example: the sequence $(1)$ is not a base of $S_3$, because the element $(2\\,3) \\ne e$ fixes $1$.\nAlso, $S = \\set{(1\\,2\\,3), (1\\,2)}$ is not a strong generating set for the base $(1, 2)$: no element of $S$ fixes $1$, so $S \\cap G^{(2)}$ generates only $\\set{e}$.\n",
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