Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem.

Sometimes, it goes the other way, i.e. algebraic topology is used to attack some problem from algebra. Here are two examples I particularly like:

1. Schreiers theorem: Every subgroup of a free group is free: This reduces to study coverings of some wedge of circles.

2. Group cohomology: E.g. questions like "Does every finite group have nontrivial cohomology in infinitely many degrees?" - see Non-vanishing of group cohomology in sufficiently high degree. Such questions can be phrased purely algebraically, but still the best way to think about them is via topology.

I am interested in seeing more examples of this kind. What is your favourite application of topology in algebra?

Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem.

Sometimes, it goes the other way, i.e. algebraic topology is used to attack some problem from algebra. Here are two examples I particularly like:

1. Schreiers theorem: Every subgroup of a free group is free: This reduces to study coverings of some wedge of circles.

2. Group cohomology: E.g. questions like "Does every finite group have nontrivial cohomology in infinitely many degrees?" - see Non-vanishing of group cohomology in sufficiently high degree. Such questions can be phrased purely algebraically, but still the best way to think about them is via topology.

I am interested in seeing more examples of this kind. What is your favourite application of topology in algebra?