Examples of groups in the real world

Question

I'm looking for some examples of groups in the real world to show students in a liberal arts math course. For example the Rubik's cube. Keep in mind these students have only a college algebra background.

Answer 1

In music, the cyclic group of order $12$, $\mathbb Z/12\mathbb Z=\{0,1,\ldots,11\}$, occurs as the group of intervals in the chromatic scale ($\mathbb Z$) modulo octave equivalence ($12\mathbb Z$). It has exactly four generators: $\pm 1$ (semitones up and down) and $\pm 7$ (perfect fifths up and down). The group acts freely and transitively on the set of $12$ pitch classes $\{C, C\sharp, \ldots, B\}$. The orbit of the generator $1$ is called the chromatic circle. The orbit of the generator $7$ is called the circle of fifths. (Both orbits are the same set, the set of all $12$ pitch classes, with a different graph structure.)

Answer 2

The group of permutations of $52$ objects can be identified with shuffling a deck of cards. Every time someone shuffles a pack they are performing a group operation.

There are some nice card tricks that involve group theory. I don't remember the details exactly, but one involves addition modulo four in relation to the suits.