Augustin Louis Cauchy

1. Augustin Louis Cauchy 1789-1857

2. Quick Facts • Born in Paris, France • Died in Sceaux, France at the age of 68 • Lived during the French Revolution • Became a military engineer and worked on the harbors and fortifications for Napoleon’s English invasion fleet

3. Quick Facts Continued • At a young age, Cauchy was made a professor at École Polytechnique In Paris, where he taught calculus. 3 Treatises • Cours d’analyse de l’École Royale Polytechnique (1821; “Courses on Analysis from the École Royale Polytechnique”) • Résumé des leçons sur le calcul infinitésimal (1823; “Résumé of Lessons on Infinitesimal Calculus”) • Leçons sur les applications du calcul infinitésimal à la géométrie (1826–28; “Lessons on the Applications of Infinitesimal Calculus to Geometry”)

4. Pioneered the study of: • Analysis (real & complex) • Theory of Permutation groups • Convergence and divergence of infinite series • Differential Equations • Determinants • Probability • Mathematical Physics

5. Awards & Honors • Fellow of the Royal Society (1832) • Fellow of the Royal Society of Edinburgh (1845) • Street name (Rue Cauchy) • Lunar Features (Crater Cauchy and Rupes Cauchy) • Commemorated on the Eiffel Tower • 16 concepts and theorems are name after him

6. Cauchy’s Theorems • Cauchy's integral theorem in complex analysis, also Cauchy's integral formula • Cauchy's mean value theorem in real analysis, an extended form of the mean value theorem • Cauchy's theorem (group theory) • Cauchy's theorem (geometry) on rigidity of convex polytopes • The Cauchy–Kovalevskaya theorem concerning partial differential equations • The Cauchy–Peano theorem in the study of ordinary differential equations

8. Cauchy’s Abelian Group Theorem If G is a finite abelian group and p is a prime that divides|G|, then ∃g ∈ G such that |g| = p.

9. Proof: (We prove this by strong induction on the order of G.) Base Step: When |G| = 2, the only prime that divides |G| is 2. Let g be a nonidentity element in G, then g² is the identity, hence |g| = 2

10. Induction Step: Now assume the theorem holds for all abelian groups of order less than n and suppose |G| = n. • Let a be any nonidentity element of G. • Then the order of a is a positive integer and is therefore divisible by some prime q (by the Fundamental Theorem of Arithmetic). • Then |a| = qt for some positive integer t. • Let b = then |b| = q.

11. Case 1: q=p Then we are done. Case 2: q≠p • Let N be cyclic subgroup <b>. • Since Gis abelian, N is normal and |N| = q. • Then |G/N| = • |G|/|N| = n/q (by Lagrange’s Theorem). • But n/q < n. • Thus, by the induction hypothesis, the theorem is true for G/N.

12. Note that |G| = |N||G/N| = q|G/N|. • Since p||G| and q≠ p, p divides |G/N|. • Thus, G/N contains an element of order p, say, Nc.

13. Note that = = Ne (where e denotes the identity of G), thus N. • Also, = = e. • Thus, c must have order dividing pq. • Note that c cannot have order 1, for otherwise Nc would have order 1 instead of p. • Also, c cannot have order q, for otherwise = Np|q, contradicting the fact that q is a prime different from p.

14. Thus, we are left with the possibilities that |c|=p or |c|=pq. • In the first case, set g=c. • If it is the second case, set g=. • Therefore, the theorem holds for abelian groups of order n, for any positive integer n.

15. Cauchy’s Theorem holds for any finite group. • An important application of Cauchy’s Theorem is that the converse of Lagrange’s Theorem holds for any finite commutative group. (Let G be a finite commutative group of order n. If m is a positive integer such that m|n then G has a subgroup of order m.)

16. Consequences of Cauchy’s Theorem • Corollary: The size of any finite field is a prime power. • Corollary: Any finite abelian group is isomorphic to a direct product of finite abelian groups with prime-power size. • Theorem: Let p,q be distinct primes, with p<q. If q1modp, then all groups of size pq are cyclic. In particular, all groups of size pq are isomorphic. If q1modp, then up to isomorphism there are two groups of size pq. • Lemma: Let G be a group of size pq, where p and q are prime with p<q. There is only one subgroup of G with size q. • Theorem: Let p,q be primes where p<q. Any abelian group of size pq is cyclic. If q1modp, any group of size pq is abelian, and thus is cyclic. • Theorem: Let f(x) be a non-constant polynomial with coefficients in Z/(p), of degree d. Then f(x) has at most d roots in Z/(p).

17. References • http://www.britannica.com/EBchecked/topic/100302/Augustin-Louis-Baron-Cauchy • http://www-history.mcs.st-and.ac.uk/Mathematicians/Cauchy.html • http://math.berkeley.edu/~robin/Cauchy/ • http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchyapp.pdf • http://johnnykwong.files.wordpress.com/2009/06/cauchy-1.pdf