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  "body": "\n## Example\n\nThe group $A_5$ has order $60$.\nIts [[conjugacy classes|0DXD4QSZ8A]] have sizes $1$, $15$, $20$, $12$ and $12$.\nThe class of size $1$ is $\\{e\\}$, and the class of size $20$ contains the $3$-cycles.\nThe class of size $15$ contains the products of two disjoint transpositions.\nThe $24$ five-cycles split into two classes of size $12$.\nA normal subgroup $N$ of $A_5$ is a union of conjugacy classes, and it contains the class $\\{e\\}$.\nBy [[Lagrange's theorem|3C53TD]], $|N|$ divides $60$.\nNo sum of $1$ and some of the numbers $15, 20, 12, 12$ divides $60$, except $1$ and $60$.\nThus $N = \\{e\\}$ or $N = A_5$, and $A_5$ is simple.\nIt is the smallest simple group that is not abelian.\n",
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