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  "body": "\n## Definition\n\nA [[group|G5S1M9]] $G$ is simple when $G \\neq \\{e\\}$ and the only [[normal subgroups|0CB0DX5BVV]] of $G$ are $\\{e\\}$ and $G$.\n\n## Remarks\n\nThe cyclic groups $\\mathbb{Z}/p\\mathbb{Z}$ of prime order $p$ are simple, and they are the only [[abelian simple groups|0JJV80RXHD]].\nThe [[alternating group|0Z5905Q73D]] $A_n$ is simple for $n \\geq 5$, and $A_5$ is the smallest simple group that is not abelian.\nA simple group has only the trivial quotient and itself as quotients, so it cannot be split into smaller pieces by a quotient.\n\n## Explanations\n\n### L1 human\n\nA simple group cannot be broken into a normal subgroup and a quotient group.\nIn this way, simple groups are like the prime numbers among the integers.\nEach finite group is built from simple groups, by the Jordan–Hölder theorem.\n",
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