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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?

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lagrange s theorem1
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 2257-1 prime?
    • Is computer security possible?
    • etc.
recall
Recall:
  • Let H be a subgroup of G, and a,b in G.
  • 3. aH = bH iff a belongs to bH
  • 4. aH and bH are either equal or disjoint
  • 6. |aH| = |bH|
lagrange s theorem2
Lagrange\'s Theorem
  • If G is finite group and H is a subgroup of G, then
    • |H| divides |G|.
    • The number of distinct left (right) cosets of H in G is |G|/|H|
my proof
My proof:
  • Let H ≤ G with |G| = n, and |H| = k.
  • Write the elements of H in row 1:
my proof1
My proof:
  • Choose any a2 in G not in row 1.
  • Write a2H in the second row.
my proof2
My proof:
  • Continue in a similar manner…
  • Since G is finite, this process will end
my proof3
My proof:
  • Rows are disjoint by (4)
  • Each row has k elements by (6)
my proof4
My proof:
  • Let r be the number of distinct cosets.
  • Clearly |G| = |H|•r, and r = |G|/|H|.
what does lagrange s theorem say
What doesLagrange\'s Theorem say?
  • Let H ≤ G where |G| = 12.

Then |H| could only be…

1, 2, 3, 4, 6, 12: The divisors of 12.

  • G =Z12 is cyclic, so there is exactly one subgroup of each of these orders.
  • G = A4 is not cyclic, and there is no subgroup of order 6.
  • The converse of Lagrange\'s theorem is False!
definition
Definition
  • Let H be a subgroup of G.
  • The number of left (right) cosets of H in G is called

the index in G of H

and is denoted |G:H|.

g h g h
|G:H| = |G|/|H|
  • Corollary 1: If G is a finite group and H is a subgroup of G, then |G:H| = |G|/|H|.
  • Proof: This is a restatement of Lagrange\'s theorem using the definition of the index in G of H.
a divides g
|a| divides |G|
  • Corollary 2. In a finite group G, the order of each element of the group divides the order of the group.
  • Proof: Let a be any element of G. Then |a| = |<a>|. By Lagrange\'s Theorem, |<a>| divides |G|.
groups of prime order
Groups of prime order
  • Corollary 3. A group of prime order is cyclic.
  • Proof: Let |G| be prime. Choose any a≠e in G. Then |<a>| > 1.

Since |<a>| divides |G|, |<a>| = |G|

It follows that G = <a>

So G is cyclic.

a g e
a|G| = e
  • Corollary 4. Let G be a finite group, and let a belong to G. Then a|G| = e.
  • Proof: By corollary 2, |a| divides |G|, so

|G| = |a|k for some positive integer k.

Hence a|G| = a|a|k = ek = e.

fermat s little theorem
Fermat\'s little theorem
  • For every integer a and every prime p,

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.

fermat s little theorem con t
Fermat\'s little theorem (con\'t)
  • In case r = 0, 0p mod p = 0.
  • If r > 0, then r in U(p) = {1, 2, …, p-1}.

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.

example find 50 11 mod 11
Example: Find 5011 mod 11
  • 5011 mod 11

= 50 mod 11 = 6

  • Check it:

5011 = 4,882,812,500,000,000,000

= 11•443,892,045,454,454,454+6

So 5011 mod 11 = 6

example 2 257 1 not prime
Example: 2257-1 not prime.
  • Suppose, towards a contradiction, that

p = 2257-1 is prime.

Using Python, we get

p = 231584178474632390847141970017375815706539969331281128078915168015826259279871

  • It is easy to calculate p, but factoring is hard!
2 257 1
2257-1
  • However 10p mod p = 10

So 10p+1 mod p should be 100.

  • To calculate 10p+1, note that
2 257 11
2257-1
  • In Python:

p = 2**257-1

t = 10

for n in range(257):

t = (t*t)%p

print t

2 257 12
2257-1
  • 23323117726701610548024580880832227821258735681932676554551014701139464992104
  • Since this number is not 100, p is not prime.
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