Group Theory/Characteristic subgroups

We conclude:

Exercises

 * 1) Prove that all subgroups of $$\mathbb Z_6$$ are characteristic.
 * 2) Let $$H, L$$ be two finite simple groups such that $$|L|$$ is divisible by a prime number $$p$$ that does not divide $$|H|$$. Use the structure theorem for characteristically simple groups to prove that $$H \times L$$ is not characteristically simple.
 * 3) Prove that a subgroup of a characteristically simple group need not be characteristically simple.
 * 4) Prove that the product of characteristically simple subgroups whose minimal normal subgroups are not isomorphic is not characteristically simple.