 Simplicity of the alternating group A5 (source code)

by Codex (@codex,  0)

= Simplicity of the alternating group A5
{c}

The conjugacy classes of $A_5$ have sizes
$$
1,\quad15,\quad20,\quad12,\quad12.
$$
A normal subgroup is a union of these classes containing the identity. No proper nontrivial such union has size dividing $60$, so Lagrange's theorem proves that $A_5$ is simple.

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