# How do I prove that all symmetries have an inverse?

I am currently studying discrete symmetries in quantum mechanics and have trouble proving that the set of discrete symmetry operators is a group. An operator, $$\hat S$$, is called a symmetry operator if the system Hamiltonian, $$\hat H$$, is invariant under the action of $$\hat S$$,

$$$$\hat S^\dagger \hat H \hat S = \hat H. \tag{1} \label{eq:1}$$$$

Most books state that the set of symmetry operators is a group without taking much time to prove it. Associativity and existence of a neutral element is easy to prove since the composition of operators is associative and the identity operator, $$\hat I$$, acts as neutral element. However, I find it hard to prove, using only equation \eqref{eq:1}, that every discrete symmetry has an inverse $$\hat S ^{-1}$$. Most resources I have checked simply claims this to be the case and then presents some examples.

How can I prove that every discrete symmetry has an inverse?

• Related: physics.stackexchange.com/q/19751/2451 and links therein. – Qmechanic Apr 25 at 14:08
• The operator $\hat S$ must also preserve $|\langle \psi_1|\psi_2\rangle|$ and so has to be unitary or antiunitary, and therefore has an inverse. – mike stone Apr 25 at 14:17
• That must be the answer! Thank you! – Michael Iversen Apr 25 at 14:21