The Bra-Ket (bracket) is primarily a notation for functions in a function space.

Group theory is applicable to the eigenfunctions of a system and there are quantum systems in which a cyclic group occurs:

Article Cyclic groups and quantum logic gates

But that's not a general statement for all systems.

By bracket we usually mean commutator.

In that case no.

What is the bracket for you?

You mean a bra and a ket together?

Still no. It means psi* psi

You mean like a ring of identical atoms?

That is solvable.