Group theory is an abstract way of mathematically analyzing symmetries. Unitary operators are paramount to quantum theory, so unitary groups are important in particle physics. The group of N dimensional unitary square matrices is denoted U(N). Unitary operators preserve inner products which means probabilities are also preserved, so the quantum mechanics of the system is invariant under unitary transformations. Let U ^ ^ be a unitary operator, so the inverse is the Hermitian adjoint U ^ ^ − − 1 = U ^ ^ † † , which commutes with the Hamiltonian: