(1) you can use $\mathbb{F}_2$ to prove that every group $G$ with $Aut(G)\cong 0$ is either $0$ or $\mathbb{Z}/2\mathbb{Z}$.
This is done by noting that the group is abelian, since all conjugation-automorphisms are the identity.
Then for abelian groups one has the automorphism $g\mapsto -g$ so all elements are self inverse.
At this point one get's gets that $G$ is a $\mathbb{F}_2$-vector space. Since any vector space of dimension $≥2$ admits nontrivial automorphisms, the result follows.
(2) also you might want to take a look at this: http://gowers.wordpress.com/2008/07/31/dimension-arguments-in-combinatorics/
