Lagrange's Theorem

1. Lagrange's Theorem

2. 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.

3. 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|

4. 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|

5. My proof: • Let H ≤ G with |G| = n, and |H| = k. • Write the elements of H in row 1:

6. My proof: • Choose any a2 in G not in row 1. • Write a2H in the second row.

7. My proof: • Continue in a similar manner… • Since G is finite, this process will end

8. My proof: • Rows are disjoint by (4) • Each row has k elements by (6)

9. My proof: • Let r be the number of distinct cosets. • Clearly |G| = |H|•r, and r = |G|/|H|.

10. 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!

11. 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|.

12. |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.

13. |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|.

14. 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.

15. 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.

16. 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.

17. 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.

18. 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

19. 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!

20. 2257-1 • However 10p mod p = 10 So 10p+1 mod p should be 100. • To calculate 10p+1, note that

21. 2257-1 • In Python: p = 2**257-1 t = 10 for n in range(257): t = (t*t)%p print t

22. 2257-1 • 23323117726701610548024580880832227821258735681932676554551014701139464992104 • Since this number is not 100, p is not prime.