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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\n$G$ is *residually finite* if for every $g \\in G$ with $g \\ne 1$ there is a [[normal subgroup|0CB0DX5BVV]] $N$ of $G$ of finite [[index|0DTXBQPA2S]] with $g \\notin N$.\n\n## Remarks\n\nEquivalently, for every $g \\ne 1$ there are a finite group $Q$ and a [[homomorphism|0JYTG5K9Y4]] $\\varphi \\colon G \\to Q$ with $\\varphi(g) \\ne 1$.\nEquivalently, the intersection of all normal subgroups of finite index of $G$ is trivial.\n\nExample: $\\Z$ is residually finite.\nFor $n \\ne 0$, the subgroup $N = (\\abs{n} + 1)\\Z$ is normal of index $\\abs{n} + 1$, and $n \\notin N$ since $0 \u003c \\abs{n} \u003c \\abs{n} + 1$.\n\nNon-example: the additive group $\\Q$ is not residually finite.\nEach quotient of $\\Q$ is divisible, and a finite divisible abelian group is trivial, since multiplication by its order is surjective and also zero.\nSo $\\Q$ has no proper normal subgroup of finite index, and the condition fails for the element $1 \\in \\Q$, which is not the identity element $0$.\n",
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