TITLE: (Elementary) applications of group (co-)homology
QUESTION [6 upvotes]: I am looking for an elementary example of a problem, for which one does not need many things to understand the question, but which can be solved with group homology or cohomology.
My background is, that I am looking for an introductional problem to motivate a talk about group homology and cohomology in a beginner's course.
Thank a lot!

REPLY [2 votes]: In DC Isaksen - ‎2002 they show that the "carry" in elementary addition can be seen as a cohomology class. 
One of my favorite application of cohomology is to classify wallpaper groups. Most wallpaper patterns are symmetric under a subgroup of all affine isometries. The question is what are all the subgroups that can arise in this way. If you look at wikipedia you'll find that there are 17 possibilities but the actual classification is messy and relies on a lot of special cases and "annoying geometry". Cohomology simplifies the picture a lot as you can see in this document from Morandi. Basically a wallpaper group is determined by a lattice subgroup (the translations by which the pattern is invariant), a point group (the isometries that leave a point fixed up to the action of the lattice group) and a cohomology class which characterizes how the wallpaper group is built up from the lattice and point groups.