How does the locally compact group play a role in many branches of math? [closed]

It seems that Fourier analysis/harmonic analysis plays an important role in math, at least in number theory and statistics. It also seems to me that talking about it is talking about locally compact groups. So to have a big picture, can you explain to me how the local and non-local compact, compact and non-compact groups play a role in many branches of math?

I'm looking for an answer similarly to the one in What does $$S^1$$ do in many branches of math? My background is Dym & McKean, Fourier Series and Integrals. I still don't understand much what that group mean though.

closed as too broad by Trevor Gunn, Lord Shark the Unknown, Namaste, hardmath, ShaileshNov 17 '18 at 0:03

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• The scope of your Question could be narrowed. Evidently you are interested not only in locally compact groups, but in topological(?) groups and their applications generally, but such a broad topic is a poor fit for the Math.SE format. – hardmath Nov 16 '18 at 17:42