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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.

Wintermute
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    Very similar to http://math.stackexchange.com/questions/65300/real-world-uses-of-algebraic-structures?rq=1 – Listing Jan 08 '14 at 18:23
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    @Listing The question you refer to is to technical for the population of students I'm considering. I looking for examples provided by familiar things you could present to the average person on the street. – Wintermute Jan 08 '14 at 18:26
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    The group of plane isometries, and the subgroup of rotations/translations. Maybe that's too obvious. – user119908 Jan 08 '14 at 18:29
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    Permutation groups (e.g., shuffling cards). – Greg Martin Jan 08 '14 at 19:01
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    The seven Frieze groups are a standard photo project for our liberal arts math class. – Jack Schmidt Jan 08 '14 at 19:09

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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.

Fly by Night
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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.)

Chris Culter
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