Group: Fermat Little Theorem

Use Group to prove Fermat Little Theorem:
For any prime p,
Let Group (Zp,*mod p) = {1,2,3....p-1};    [*mod p= multiply modulo p]
For any non-zero b in Zp,
$latex b^{p-1} = 1 \: \mbox {in Zp }$
Since Zp isomorphic~ to the ring of cosets  of the form a+pZ   (eg. Z2 ~ {0+2Z, 1+2Z}
For any m in Z not in the coset {0+pZ}
ie m ≠0 (mod p)
or p not divisible by a
$latex m^{p-1} = 1 (mod \: p)$
(x m both sides)
$latex m^{p} = m (mod \: p)$