ElGamal digital signature. why mod p-1 in delta?

Im trying to figure out why $\delta$ in the signature part of ElGamal has mod $p-1$ when $\gamma$ has mod $p$?

$\gamma = \alpha^k \mod p$

$\delta = (x-a\gamma)k^{-1} \mod p-1$

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Basically because of Fermat's little theorem: if $a$ is not divisible by $p$ then $a^{p-1} = 1$ $mod$ $p$. A part of the expression for $\delta$ appears as a power of $a$ in the ElGamal signature verification equation, which "happens" to work because it is reduced modulo $p-1$ so Fermat's little theorem applies.
Why can you when proving the verification, raise gamma when it is mod $p$? $\alpha^x = \beta^\gamma \gamma \delta \mod p$ – Becktor May 19 '14 at 16:53
What's really going on is that $\beta$ and $\gamma$ are both themselves powers of $\alpha$. You need to expand them to see what's going on. – otus May 19 '14 at 17:07
Yea I $a^x=\beta^\gamma \gamma^\delta$ and get: $\alpha^{a\gamma}\alpha^{k(x-a\gamma)k^{-1}}$ which gives $\alpha^{a\gamma}\alpha^{x}\alpha^{-a\gamma}$, So since $\alpha^{a\gamma}$ and $\alpha^{-a\gamma}$ cancels each other we can do this? – Becktor May 19 '14 at 17:15
That's it, basically. You need the $mod$ $p-1$ so that the power is "off" by a multiple a $p-1$ (which by F.l.t. means the power is one). Otherwise you couldn't cancel the $k$'s. – otus May 19 '14 at 17:19