Prove that if $g^{(p-1)/2} \equiv -1 (mod \mbox{ } p)$ then its smaller power can't be congruent to 1.

Question

Let $p$ be a prime greater than $2$. Show that $g^{(p-1)/2} \equiv -1 (mod \mbox{ }p)$ implies $g^{k} \not\equiv 1 (mod \mbox{ }p)$ for every $1≤k≤(p-1)/2$.

Answer 1

If $g^{(p-1)/2}\equiv-1\pmod p$

using Discrete Logarithm wrt primitive root $a,$

$\dfrac{p-1}2\cdot$ind$_ag\equiv$ind$(-1)\pmod{\phi(p)}$

As ind$(-1)\equiv\dfrac{p-1}2\pmod{\phi(p)}$

$\dfrac{p-1}2($ind$_ag-1)\equiv0\pmod{p-1}$

$\implies$ind$_ag$ must be odd.

Now from this, if $d|(p-1),$ there exist $\phi(d)$ values of $g$ such that ord$_pg=d$

As ind$_ag\mid(p-1),$ there will be $\phi($ind$_ag)$ values of $g$