Dear Augustin Cauchy,
Your mathematical accomplishments have had an enormous impact in the fields of math and science. More concepts and theorems have been named after you than any other mathematician alone, and to this day, you have written the second largest number of papers of any mathematician, only second to Leonard Euler. Some areas you have had influence in are complex functions, optics, elasticity, group theory, mathematical physics and astronomy, hydrodynamics, and differential equations. Even though you are not most well-known for your work in algebra, I first heard of you in an Abstract Algebra course. In the field of algebra, your early work is the basis of group theory, and you have contributed to finding the inverse of a matrix, creating theorems on determinants formed by subdeterminants, and you proved a generalization of one of Ruffini's theorems. Even though it took some time after you for abstract group theory to form and for it to move alway from the term substitution, which is your word for permutation, your work was still fundamental. Algebra is one of many fields in which you have a theorem named after you. Your theorem for finite groups, which is called Cauchy's Theorem, states, "Let G be a finite group and p a prime such that p divides the order of G. Then G has an element of order p." It implies that if a prime number divides the order of a group, then that group has a subgroup whose order is p, which is the cyclic group generated by p. This proof can be done using strong induction, the class equation, and group actions. Your original proof uses complex computations involving permutation group theory and is several pages long. I will show a more concise version using induction, the class equation, and group actions:
Let G be a finite group of order n greater than or equal to p. By hypothesis, p | n. If n = p, then this theorem is trivial because for a group to be a certain order, it must have an element of that order. We proceed inductively. Suppose the theorem holds for all groups of order k such that p <= k < n. By the class equation, |G| = |Z(G)| + [G : C(x1)] + ... + [G : C(xk)], where |Z(G)| is the order of the center of G and [G : C(x1)] + ... + [G : C(xk)] are the cosets of the centralizer subgroup C(xi) of xi in G. By Lagrange's Theorem, |G| = |C(xi)|[G : C(xi)] for each i. So, for each i, either p divides |C(xi)| or p divides [G : C(xi)]. If p divides some C(xi), then we are done because |C(xi)| must be less than |G|. If p does not divide any C(xi), then p divides each [G : C(xi)]. Because p divides |G|, and every [G : C(xi)], p must also divide |Z(G)| by the rules of algebra (since |G| = |Z(G)| + [G : C(x1)] + ... + [G : C(xk)]). Hence, it follows by the Fundamental Theorem of Finite Abelian Groups that if G is a finite group and p is a prime such that p divides the order of G, then G has an element of order p.
As you can see, the study of permutation groups has progressed tremendously since you originally proved the above theorem. Cauchy's theorem was later generalized by Sylow's First Theorem, which states, "Let G be a finite group, and p a prime such that p^r divides the order of G. Then G has an element of order p^r. In my next letter, I will talk of other great contributions you have had to the development of mathematics.
Regan Kapalko
Cauchy on Permutations and the Origin of Group Theory. (n.d.). Retrieved March 13, 2018, from http://nonagon.org/ExLibris/cauchy-permutations-origin-group-theory
Chaitesipaseut, S. (n.d.). Retrieved March 13, 2018, from https://math.berkeley.edu/~robin/Cauchy/index.html
Charles Scribner's Sons. (2018). Cauchy, Augustin-Louis. Retrieved March 13, 2018, from https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/augustin-louis-cauchy
Your mathematical accomplishments have had an enormous impact in the fields of math and science. More concepts and theorems have been named after you than any other mathematician alone, and to this day, you have written the second largest number of papers of any mathematician, only second to Leonard Euler. Some areas you have had influence in are complex functions, optics, elasticity, group theory, mathematical physics and astronomy, hydrodynamics, and differential equations. Even though you are not most well-known for your work in algebra, I first heard of you in an Abstract Algebra course. In the field of algebra, your early work is the basis of group theory, and you have contributed to finding the inverse of a matrix, creating theorems on determinants formed by subdeterminants, and you proved a generalization of one of Ruffini's theorems. Even though it took some time after you for abstract group theory to form and for it to move alway from the term substitution, which is your word for permutation, your work was still fundamental. Algebra is one of many fields in which you have a theorem named after you. Your theorem for finite groups, which is called Cauchy's Theorem, states, "Let G be a finite group and p a prime such that p divides the order of G. Then G has an element of order p." It implies that if a prime number divides the order of a group, then that group has a subgroup whose order is p, which is the cyclic group generated by p. This proof can be done using strong induction, the class equation, and group actions. Your original proof uses complex computations involving permutation group theory and is several pages long. I will show a more concise version using induction, the class equation, and group actions:Let G be a finite group of order n greater than or equal to p. By hypothesis, p | n. If n = p, then this theorem is trivial because for a group to be a certain order, it must have an element of that order. We proceed inductively. Suppose the theorem holds for all groups of order k such that p <= k < n. By the class equation, |G| = |Z(G)| + [G : C(x1)] + ... + [G : C(xk)], where |Z(G)| is the order of the center of G and [G : C(x1)] + ... + [G : C(xk)] are the cosets of the centralizer subgroup C(xi) of xi in G. By Lagrange's Theorem, |G| = |C(xi)|[G : C(xi)] for each i. So, for each i, either p divides |C(xi)| or p divides [G : C(xi)]. If p divides some C(xi), then we are done because |C(xi)| must be less than |G|. If p does not divide any C(xi), then p divides each [G : C(xi)]. Because p divides |G|, and every [G : C(xi)], p must also divide |Z(G)| by the rules of algebra (since |G| = |Z(G)| + [G : C(x1)] + ... + [G : C(xk)]). Hence, it follows by the Fundamental Theorem of Finite Abelian Groups that if G is a finite group and p is a prime such that p divides the order of G, then G has an element of order p.
As you can see, the study of permutation groups has progressed tremendously since you originally proved the above theorem. Cauchy's theorem was later generalized by Sylow's First Theorem, which states, "Let G be a finite group, and p a prime such that p^r divides the order of G. Then G has an element of order p^r. In my next letter, I will talk of other great contributions you have had to the development of mathematics.
Regan Kapalko
Cauchy on Permutations and the Origin of Group Theory. (n.d.). Retrieved March 13, 2018, from http://nonagon.org/ExLibris/cauchy-permutations-origin-group-theory
Chaitesipaseut, S. (n.d.). Retrieved March 13, 2018, from https://math.berkeley.edu/~robin/Cauchy/index.html
Charles Scribner's Sons. (2018). Cauchy, Augustin-Louis. Retrieved March 13, 2018, from https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/augustin-louis-cauchy
One typo: "to move alway from the term"
ReplyDeleteIn this line, "Even though it took some time after you for abstract group theory to form and for it to move alway from the term substitution..." I meant "Even though it took some time after you for abstract group theory to form and for (abstract group theory) to move alway from the term substitution...", so I don't think I have a typo, but I didn't say what I meant as clearly as I could.
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