Math Help - primitive root

1. primitive root

Assume that the prime $p > 2$ has an odd primitive root $g$. Prove that $2p$ must have a primitive root.

2. Originally Posted by dori1123
Assume that the prime $p > 2$ has an odd primitive root $g$. Prove that $2p$ must have a primitive root.
for any prime, $2p$ always has a primitive root because if p = 2, then $U(\mathbb{Z}/4\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$ and if p > 2, then $U(\mathbb{Z}/2p\mathbb{Z}) \cong U(\mathbb{Z}/p\mathbb{Z}) \cong \mathbb{Z}/(p-1)\mathbb{Z}.$

[ $U(\mathbb{Z}/n\mathbb{Z})$ here is the (multiplicative) group of units modulo $n.$]