Why $\alpha$ in ElGamal cryptosystem has to be a primitive root?
Everything works, except the key and signature space, reduced greatly;


  The result is that the signature space (when the key is used for     signing) or the public key space (when the key is used for encryption)     may be greatly reduced from its expected size of log(p) bits, possibly     down to 1 bit (the worst case if the order of g is 2)1.


It not hard to detect a $g$ is the generator or not.