Lagrange s theorem
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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 Theorem

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Lagrange s theorem

Lagrange's Theorem


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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