In finite group theory, one uses induction heavily. Namely, you just propose a conjecture and then take a minimal counter example, i.e. a group $G$ with minimal order that does satisfy the assumptions but not the result.
For example, assume that one wants to prove that every finite group of odd order is solvable. You can assume that $G$ is a finite group of odd order which is not solvable, and every group with order odd and less than $|G|$ is solvable. Then you can deduce that $G=G'$, the group is perfect, since if $G'< G$ then $G'$ is solvable which implies $G$ is solvable.
Now you can observe that the induction is a crucial method in finite group theory. However, one cannot use induction for infinite groups. An infinite group of minimal order clearly does not make sense (I guess?). I wonder what are the main tools in infinite group theory.