I Group theory

Maybe start to tell them that mathematics in general has a huge amount of applications, even if they are not noticed at the first sight. You might want to talk about binary base.

 

Rubik's cube, wallpaper designs, solutions of various games, chemistry, Klein geometries...

 

To the determination of molecular or crystal structure with spectroscopic and diffraction methods the Group Theory is crucial.
In case of a molecule, its symmetry elements, rotations and reflections determine its "point group", and the number of the absorption bands in the infrared or Raman spectrum can be determined from the properties of the point group. Examining the bands help to decide about the symmertry of the molecule.
Similarly, the symmetry of a crystal (its space group) can be deduced from the X-ray diffraction pattern, also using Group Theory.

 

There's a difference between asserting the importance of group theory and showing it !

It's easy to assert the importance of group theory in physics, but I've never seen an example that showed it and didn't also require a considerable amount of physics. (As an amusing example, there are various YouTube videos about finding the "point groups" of molecules. To a mathematician, but non-physicist, the terminology is confusing. The topic, from a mathematical point of view, appears to be "group actions" ).

Most students can get interested in simple abstract concepts, even if examples of applying the concept are somewhat abstract. Finite groups can always be introduced as set of functions that permute a set of letters - no need to talk about permuting atoms. (Since students may have been drilled to think that a "permutation" is "an arrangement" rather than "a process of arranging", it would be useful to get the more general meaning of "permutation" into their heads early in their careers.) Permutation groups can be represented a permutation matrices - which could be a nice way to introduce matrices. It also makes clear the distinction between "the process of arranging" (e.g. the result of a product of matrices) and "an arrangement" (e.g. the result of a matrix times a column vector).

An expert on ciphers might provide ways to show permutation groups applied to coding and decoding.

 

Much of Robert Fripp's music is applied elementary group theory, as far as I'm concerned.