I was wondering if there are any applications of abstract algebra to engineering and where I can go to learn about them?
There are some applications of group theory to problems with complcated symmetries.
For example, there are vibration patterns of a circular disk supported at the center, with N radial and M cirumferential nodal lines, for any integers N >= 0 and M >= 0. That's simple enough.
Now consider the vibration patterns of something like a fan with K identical blades mounted on a central hub. There are analogous sets vibration patterns, but they are not so obvious because every blade may be vibrating in a different way, for example if there are no common factors between the number of blades and the number of radial "nodal lines".
Now consider the case where these vibration patterns are rotating around the disk as travelling waves, rather than forming a "stationary" vibration pattern"
Now consider the disk is also rotatiing, but not at the same speed as the waves are rotating around it.
And finally couple several of these together one flexible rotor, with different numbers of blades in each disk...
Some general theory of to how to keep track of what is going on is quite useful here
It gets even more interesting when you make the more realistic assumption that the blades are only approximately identical, to within manufacturing tolerances etc.