= Simplicity of the alternating group on six letters
A <normal subgroup> of $A_6$ has order dividing 360 and consists of the identity plus whole <conjugacy classes of the alternating group on six letters>. If it omits the odd-sized class of size 45, its order is an odd divisor and no nontrivial class sum fits. If it includes that class, the possible class sums not exceeding 180 are $46,86,118,126,136,158,176$, none dividing 360. Every proper subgroup has order at most 180; therefore $A_6$ is a <simple group>.
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