Lagrange's Theorem. Lagrange's Theorem. The most important single theorem in group theory. It helps answer: How large is the symmetry group of a volleyball? A soccer ball? How many groups of order 2p where p is prime? (4, 6, 10, 14, 22, 26, …) Is 2 257 -1 prime?
Then |H| could only be…
1, 2, 3, 4, 6, 12: The divisors of 12.
the index in G of H
and is denoted |G:H|.
Since |<a>| divides |G|, |<a>| = |G|
It follows that G = <a>
So G is cyclic.
|G| = |a|k for some positive integer k.
Hence a|G| = a|a|k = ek = e.
ap mod p = a mod p.
Proof: To simplify notation, Let a mod p = r.
Then ap mod p = (a mod p)p mod p = rp mod p.
It remains to show that
rp mod p = r
for 0 ≤ r < p.
By corollary 4, r|U(p)| = rp-1 = 1 in U(p).
In other words, rp-1 mod p = 1.
So, rp mod p = r.
= 50 mod 11 = 6
5011 = 4,882,812,500,000,000,000
So 5011 mod 11 = 6
p = 2257-1 is prime.
Using Python, we get
p = 231584178474632390847141970017375815706539969331281128078915168015826259279871
So 10p+1 mod p should be 100.
p = 2**257-1
t = 10
for n in range(257):
t = (t*t)%p