Congruence in the Schnorr identification scheme

I have been looking at the Cryptography: Theory and Practice book by Stinson and Paterson and when I came to the Schnorr identification scheme, I read the sentence that goes something like this:

Observe that $$v$$ can be computed as $$(\alpha ^a)^{-1} \bmod p$$, or (more efficiently) as $$\alpha ^{q-a}\bmod p$$.

In this context $$\alpha$$ is an element having prime order $$q$$ in the group $$\mathbb{Z}_p^*$$ (where $$p$$ is prime and $$q\mid p-1$$), $$a$$ is a private key ($$0\leq a\leq q-1$$), and $$v$$ is a public key, constructed as $$v=\alpha ^{-a} \bmod p$$.

My question is, how can we get $$\alpha ^{q-a}\bmod p$$ from $$(\alpha ^a)^{-1} \bmod p$$?

Note that in any group, the exponent is computed modulo the order of the group. Thus $$\alpha^{-a} = \alpha^{-a \bmod q} = \alpha^{q-a}$$.