This problem might be related to some of the ones suggested above, I did not check carefully.
Assume that you have to prepare the calendar of a basketball league to which $2n$ teams take part, where $n$ is an integer $\geq 1$. The full season consists of $2n-1$ dates where in each date each of the team has to play against another one (different than itself :-)). This has to be done in such a way that, overall, every team meets every other team exactly once (we all know what the calendar of a basketball league look like).
The preparation of the calendar can easily be accomplished using some very very basic group theory as follow. Let $G$ be the group ${(\mathbf{Z}/2\mathbf{Z})}^n$ and define the set of dates of the calendar to be $G$ deprived by the identity element. Identify now the $2n$ teams of the league with $G$ by picking any bijection. Then the typical game of the calendar on the date $D$ will look like ($X$ vs $X+D$).
I learnt about the idea of using group theory to solve the "calendar problem" long time ago from a friend who attended the Scuola Normale Superiore in Pisa. I have the impression that the solution he explained to me then did not involve the particular group $G$ given above, but instead an arbitrary abelian group of order $2n-1$ parametrizing the dates of the calendar. Unfortunately that solution, more elegant but perhaps less practical (!), does not occur to me at the moment.