If you love groups, you'd love my job. My job is to study Coxeter groups (and some of their cousins, e.g. Artin groups).
Non-abelian groups can be a pain, but I think they're also the real reason why we invented groups in the first place - to deal with things like permutations and functions that simply don't commute. If all we ever dealt with was abelian groups, we probably wouldn't have come up with beautiful ideas like the Sylow theorems, and we certainly wouldn't be able to talk about groups acting on geometric cell complexes (which is my personal meal ticket
).
What is your Job, eh? I personally am fond of Lie Groups which are continuous.