# What are the most important implications of group theory?

I am just starting to investigate group theory and wonder why it is seen as so important in the mathematical world. Certainly it seems that it is very flexible in its uses and we continually hear of Galois theory, but what else is there?

Please reply in a fashion that can be understood by someone who, although knowing the basics of group theory, does not know of any particularly important groups (but would be very interested in learning of them)

• As a physicist, I think Noether's theorem, which relates symmetry groups to physically conserved quantities, is pretty important. – user137731 Dec 8 '14 at 15:04

Often when groups act on an object, you can learn a lot about the object itself from studying the group action. Also, the orbit space often has interesting structure. A very basic example of this would be the torus as the orbit space of $\mathbb{R}^2/\mathbb{Z}^2$.