I think a nice way to introduce groups is the Rubik's Cube non-commutative group. Also tessellation can be a way to inspire the need of a simple underlying structure to represent some complex sets. Also the dihedral group has the advantage of being visual.

As dtldarek pointed out, you can praise finite fields for their applications in asymmetric cryptography (RSA) or error-correcting codes like Reed-Solomon with their application to CDs. Or more generally the existence and ease of computing the inverse of any element of a finite field using the extended Euclidean algorithm.

My favorite hidden application of algebra is the Spot It! game, which uses projective planes over finite fields.

I think a nice way to introduce groups is the Rubik's Cube non-commutative group. Also tessellation can be a way to inspire the need of a simple underlying structure to represent some complex sets. Also the dihedral group has the advantage of being visual.

As dtldarek pointed out, you can praise finite fields for their applications in asymmetric cryptography (RSA) or error-correcting codes like Reed-Solomon with their application to CDs. Or more generally the existence and ease of computing the inverse of any element of a finite field using the extended Euclidean algorithm.

My favorite hidden application of algebra is the Spot It! game, which uses projective planes over finite fields.