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 m in Zp,
m^{p-1} = 1 \: \mbox { in Zp }
Since Zp isomorphic~ to the ring of co-sets  of the form m+pZ   (eg. Z2 ~ {0+2Z, 1+2Z}
For any m in Z not in the co-set {0+pZ}
ie m ≠0 (mod p)
or p not divisible by m
m^{(p-1)} \equiv 1 \mod p, \forall m \text{ coprime p}

For general case: no need the (m, p) co-prime condition

(x m both sides)

\boxed{    m^{p} \equiv m \mod p, \forall m }

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