>>10985843 is a group.
Anyway, groups are very useful for having something of a "minimal" structure while still being very powerful tools for proving properties about classes of objects that might not be under control otherwise. I'm not mathematician (I do physics instead), but even for my "applied" purposes it's very useful to work with groups or to know that the objects I work with are groups. Obviously physics uses Lie groups constantly, as they're the backbone of describing the dynamics of states on a Hilbert space. They also provide gauge degrees of freedom that high energy physicists are constantly interested in (I'm not one so I'm not an expert in this usage).
Personally I work a lot with the former notion, and even when you're not working with Lie groups specifically, group structure is still invaluable. For instance, the Clifford group is a very important group of quantum operations, and just by knowing whether an operation is Clifford or not (even if I don't have a specific construction of the unitary), I can draw some very important conclusions about my system (see: Gottesmann-Knill theorem). And how do I know whether I have a Clifford operation or not? And it's typically easy to check whether you have a Clifford operation or not: since it's a group, a Clifford is just a product of other Cliffords, and I can build up to this complicated Clifford by only thinking about simpler Cliffords. Therein lies the power of the group structure.
Also, group homomorphisms (esp. isomorphisms) are invaluable. God forbid I know what the hell is going on with the actual group elements to form an -fold symmetry, but fortunately I just need to understand and the first isomorphism theorem to work it all out.
