Real-world uses of Algebraic Structures

Solution 1:

Here's one place to start. The Unreasonable Effectiveness of Number Theory contains the following interesting surveys. Their references should provide good entry points to related literature.

• M. R. Schroeder -- The unreasonable effectiveness of number theory in physics, communication, and music

• G. E. Andrews -- The reasonable and unreasonable effectiveness of number theory in statistical mechanics

• J. C. Lagarias -- Number theory and dynamical systems

• G. Marsaglia -- The mathematics of random number generators

• V. Pless -- Cyclotomy and cyclic codes

• M. D. McIlroy -- Number theory in computer graphics

Solution 2:

Group theory can be seen as at least one way to tackle the idea of symmetry.

For example, take something nice and symmetric like a circle (for the sake of argument, let's only consider rotational symmetries). What do you end up with?

First, you have 'actions' you can take on the circle which preserve the symmetry, for example rotating it by $\pi / 6$ radians. Second, you have the set of points of the circle itself, and third you have a way of combining them, IE a rotation of $\pi / 6$ with a starting point of $(1,0)$ gives you and ending point $(\frac{\sqrt{2}}{2},\frac{1}{2})$.

Now this is a mathematical example, but essentially any symmetry in our natural or constructed world will have something like this going on... this being called a 'group action'.