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.

  • $\begingroup$ This is an observation from afar: Whenever things get infinite, topology and measure theory seem to come into play. Functional analysis is linear algebra enhanced by topology and measure theory. For group theory, I believe it’s the same. $\endgroup$ – k.stm Dec 25 '16 at 7:33
  • $\begingroup$ Except in algebra we do not do that, it is more than analysis, measure theory/topology, important things from algebra and apply their stuff but when you remain in algebra you rarely if ever use those theories. $\endgroup$ – Zelos Malum Dec 25 '16 at 8:09
  • $\begingroup$ @ZelosMalum, allow me to disagree. If you want to study the Galois theory of infinite-dimensional field extensions, you are much better off if you introduce the Krull topology and profinite groups. $\endgroup$ – Andreas Caranti Dec 25 '16 at 11:19
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    $\begingroup$ Studying groups in complete generality is hopeless, so what tends to happen is you restrict yourself to some class of groups (finite,abelian,nilpotent,hyperbolic,amenable etc) and different techniques grow out of that. $\endgroup$ – Paul Plummer Dec 25 '16 at 15:29

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