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.