Sylow Sesquicentennial
Celebrating the 150th anniversary of the publication of Sylow’s Theorems
Peter Ludwig Sylow
(December 12, 1832-September 7, 1918)
By Robert “Dr. Bob” Gardner, Fall 2022
The Sylow Theorems are covered in Fraleigh’s A First Course in Abstract Algebra, 7th Edition (the text used in our cross-listed Introduction to Modern Algebra 2, MATH 4137/5137) in Chapter VII, “Advanced Group Theory,” Section 36 “Sylow Theorems.”
They are also covered (in a very similar fashion) in Hungerford’s Algebra (the Springer-Verlag Graduate Texts in Mathematics book which we use in the Modern Algebra 1, MATH 5410) in Chapter II, “The Structure of Groups,” Section II.5, “The Sylow Theorems.”
Peter Ludwig Mejdell Sylow
(December 12, 1832-September 7, 1918)
Christiania/Oslo, Norway
Sylow’s Stomping Grounds
Christiania (Oslo) Cathedral School
Christiania University (University of Oslo)
Niels Henrik Abel
August 5, 1802-April 6, 1829
Sylow’s Education
Carl Bjerknes
October 24, 1825-March 20, 1903
Ole Jacob Broch
January 4, 1818-February 5, 1889
Evariste Galois
October 25, 1811
-May 31, 1832
Sylow’s Inspiration
Niels H. Abel
August 5, 1802-April 6, 1829
Joseph Liouville
March 24, 1809- September 8, 1882
https://www.infoplease.com/atlas/europe (9/9/2022)
Christiania
Paris
Berlin
Sylow’s Travel
1864, The University of Christiania
Section 56. Insolvability of the Quintic
Section 53. Galois Theory Section 54. Illustrations of Galois Theory
Sylow’s Collaboration
Julius Petersen
June 16, 1839-August 5, 1910
Sophus Lie
December 17, 1842-February 18, 1899
Sylow’s University Job
1883
Göttingen Academy of Sciences and Humanities FaceBook page (9/10/2022)
1884
1894
Christiania University (University of Oslo)
Honorary Doctorate
Sylow’s Personal Life
“He never married but was a warm person with a nice sense of humour. He was an avid lover of being out of doors and often spent summer vacations in the mountains, usually in Kongsvoll, where he studied plants. Kongsvoll is a mountain station providing food and shelter on the route between Oslo and Trondheim, erected when the route was used by pilgrims visiting the shrine of St Olav in Trondheim.”
L. Sylow, ̏ Theorems on Substitution Groups,̋ Mathematische Annalen, 5(4), 584-594 (December 1, 1872).
Sylow’s Paper
150th Anniversary is December 1, 2022
Groups
Binary Operation
Groups
Group Examples 1
Group Examples 2
Group Examples 3
Group Theory History
Groups became part of algebra in the 19th century. While looking for an algebraic formula for the zeros of an nth degree polynomial (like a quadratic equation for an nth degree polynomial), Abel showed that there is not (in general) an algebraic solution to a 5th degree polynomial equation. Galois gave conditions for the existence of an algebraic solution of a general nth degree polynomial equation. These conditions involved permutations of the zeros of the polynomial.
Niels Henrik Abel
(1802-1829)
Evariste Galois
(1811-1832)
Group Theory History
Evariste Galois
(October 25, 1811-May 31, 1832)
Subgroups and Cosets
Subgroups
Cosets
Cosets
1
1
1
1
0
0
0
0
2
2
2
2
3
3
3
3
Cosets
“Definition.” When the cosets of a subgroup themselves form a group, the subgroup is a normal subgroup. The group of cosets is a quotient group (or factor group).
Definition. A group with no normal subgroups (and hence a group that cannot be used to form a quotient group) is a simple group.
Cosets
Note. Recall from Mathematical Reasoning (MATH 3000) that the equivalence classes of an equivalence relation on a set partition the set.
Cosets
Note. We now have the equipment to prove one of the first results in finite group theory.
Note. One can show that all cosets are the same cardinality.
Lagrange’s Theorem
Joseph-Louis Lagrange
(January 25, 1736-April 10, 1813)
Lagrange’s Theorem
Lagrange’s Theorem History
Camille Jordan proved Lagrange’s Theorem for the case of any permutation group in 1861. Every group is a group of permutations, so this is the general result.
So when does a group have a subgroup of a given order?
Cauchy’s Theorem
Augustin Louis Cauchy
(August 21, 1789-May 23, 1857)
Cauchy’s Theorem History
M. Meo’s “The Mathematical Life of Cauchy’s Group Theorem,” Historia Mathematica, 31 (2004), 196–221 states: “Cauchy’s theorem on the order of finite groups is a fixture of elementary course work in abstract algebra today… The initial proof by Cauchy, however, was unprecedented in its complex computations involving permutational group theory and contained an egregious error… [Cauchy’s theorem in its] most succinct form employs just the structure lacking in Cauchy’s original proof—the wreath product [i.e., group action].”
Cauchy presented his theorem in a 101 page paper in 1845.
Sylow Theorems
First Sylow Theorem
From page 209 in Section 9.4, “Sylow Theorems,” of Visual Group Theory.
Second Sylow Theorem
Third Sylow Theorem
From Robert A. Wilson’s webpage:
https://webspace.maths.qmul.ac.uk/r.a.wilson/pubs_files/Sylow.pdf
Sylow’s Paper
Sylow Theorem Example
Example 36.13. To illustrate the Sylow Theorems, we consider the following application. We claim that no group of order 15 is simple.
Sylow Theorem Example (continued)
Example 36.13. No group of order 15 is simple.
Additional Applications of the Sylow Theorems
An Application of the First Sylow Theorem
Another Application of the First Sylow Theorem
An Application of the First and Third Sylow Theorems
An Application (continued)
Example 37.10. Theorem 37.3 allows us to classify some finite groups as cyclic:
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Questions?
References
Websites