Simple group

A nontrivial simple group has no normal subgroups other than the identity subgroup and itself.

For a faithful primitive group action, suppose a point stabilizer has an abelian normal subgroup whose conjugates generate the whole group. Every nontrivial normal subgroup then contains the derived subgroup. Indeed the normal subgroup is transitive, and all those conjugates have the same image in the quotient, making the quotient abelian. If the whole group is nontrivial and perfect, it is simple.

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.

The alternating group $A_n$ is simple for every $n\ge 5$. In particular it has no proper subgroup of index two, since every index-two subgroup is normal.

A finite nonabelian simple group is a finite group that is a nonabelian group and a simple group. Every such group is a perfect group because its commutator subgroup is a nontrivial normal subgroup.

The conjugacy classes of $A_5$ have sizesA 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.

Every finite nonabelian simple group has order at least 60; the alternating group A5 attains this bound.