Prove the following statements directly (from definitions, without
using any Theorems), for a prime
p and group Z
_{p}^{*
}:
a) Exponents work mod p-1: if x
= y (mod p-1) then for
any a, a^{x }= a^{y}
(mod p).
b) If g is a generator, then g^{x }=
1 iff (p-1)|x.
c) If a is a generator, then the converse of a) also
holds: for any generator g and any x,y, if g^{x}
= g^{y}
(mod p), then x =
y (mod p-1).