Sylow and his Theorems
Michael Weiss
Upper Level Math Writing
Professor Haessig
May 5, 2007
Mission: To examine the life of Peter Sylow and to gain an understanding of his theorems through the use of explanations and applications.
Table of Contents:
- Biography of Sylow
- Necessary Definitions
- The Three Sylow Theorems
- Applications of the Theorems
- Conclusions
Biography of Peter Ludwig Mejdell Sylow
The famous Norwegian mathematician Peter Sylow was born in Christiana (now Oslo), Norway on December 12, 1832[*]. Throughout his childhood, Sylow always harbored an interest and aptitude for math and science. After completion of high school, Sylow enrolled himself at Christiana University in Oslo where he specialized in advanced mathematics. He received a great confidence booster when in 1853, he entered into a math contest and subsequently won. In 1856, although his ultimate desire was to teach at the university level, wanting to keep all his options open, he registered for and passed the Norwegian high school teacher examination. This proved to be a wise decision, as when Sylow began to search for jobs he found that no University positions were open. Consequently, Sylow ended up taking a position as a high school teacher in Frederikshald, Norway, where he would remain for 40 years teaching mathematics.
Sylow took his career to the next step beginning in 1862, when he began to substitute at Christiana University lecturing in particular on Galois Theory. (Galois Theory consists of a set of concepts that combine elements of group and field theory, more specifically it allows field theory to be reduced to group theory as to allow for less complicated problems. Évariste Galois himself was a French mathematician from Bourg-la-Reine who lived from 1811-1832.) While teaching both at Christiana University and at the high school level, Sylow continued his studies and pursued his own interests. In particular, Sylow initially concerned himself greatly with elliptical functions, however, after finding particularly interesting the question of solving algebraic equations by radicals, his focus switched to the work of Galois, who previously worked extensively on the field.
This peaked interest may be credited to a scholarship that Sylow received in 1861 to travel to Berlin, Germany and Paris, France to participate in mathematics conferences. While in Paris, Sylow attended the lectures of Michelle Chasles ((1793-1880), a French mathematician who pioneered the theories of enumerative geometry), Joseph Liouville ((1809-1882), a French mathematician who worked on number theory, complex analysis, differential geometry and topology) and Jean Marie Constant Duhamel ((1797-1872), a French mathematician who worked primarily on partial differential equations). In Berlin, Sylow was only able to attend the lecture of Leopold Kronecker (a German mathematician who worked on the theories of equations, especially in the fields of elliptical functions and the theory of algebraic functions). At the end of his travels, Sylow was known to comment that from these experiences he had ascertained an increased knowledge of the theories of equations.
Once again, in 1862, Sylow was given the opportunity to fill in for Ole Jacob Broch, yet another famous Norwegian mathematician, where he took the opportunity to discuss algebraic equations as presented through the work of Niels Henrik Abel and Évariste Galois. It is here that the foundations of the Sylow Theorems was laid when he posed the following question to himself and his students: A group of order divisible by a prime p has a subgroup of order p, can this be generalized to powers of p? It is from this question that the work of Sylow began to take off.
In 1872, Sylow published his collection of research, “Theorems sur les groupes de substitucions” in the German journal, Mathematische Annalen. It was in this journal that the three Sylow theorems detailed later in the paper were first published. In 1887, Sylow’s Theorems were proven for abstract groups by Ferdinand Frobenius, as Sylow himself had only proved them for permutation groups. These subsequent theorems would later become a foundation for almost all work in finite groups.
Later on in life, Sylow began to take more senior positions in the academic world, as demonstrated by his becoming editor of Acta Mathematica, a mathematics journal. In 1894, Sylow was awarded an honorary doctorate in mathematics by the University of Copenhagen. In 1898, he was given the position he always desired, when by decree of Sophus Lie, a special position was created for him at the University of Christiana, where he became a full professor. Peter Sylow died on September 7, 1918, a satisfied mathematician.
Definitions and Theorems
In order to effectively understand the theorems as set forth by Sylow it will be necessary to become familiar with the following definitions as taken from Section 36 of A First Course in Abstract Algebra. In addition, with some of the given definitions I have included explanations of the definitions. In the following definitions, G will refer to a group and H will refer to a subgroup of G.[†]
- Let p be a prime. A group G is a Sylow p-group if every element in G has order a power of the prime p. A subgroup of a group G is a p-subgroup of G if the subgroup is itself a p-group.
- Let p be a prime. Let G be a finite group and let p divide order(G). Then G has an element of order p and, consequently, a subgroup of order p.
- This is Cauchy’s Theorem
- Let G be a finite group. Then G is a p-group if and only if order(G) is a power of p.
- Thus, if p divides the index of G with H, then the normalizer of H is not equal to the subgroup H itself.
- Let ig: G G whereig(x) = gxg-1 for all x G be the innermost automorphism of G by g. Thus the subgroup K of G is referred to as a conjugate subgroup of H if K = ig[H] for some g G.
The Three Sylow Theorems
Now it is time to move on the actual theorems of Sylow:
First Sylow Theorem[‡]
Let G be a finite group and let order(G) = (pn)m where n>=1 and where p does not divide m. Then:
- G contains a subgroup of order pi for each i where 1<= i <= n.
- Every subgroup H of G of order pi is a normal subgroup of a subgroup of order pi+1 for 1 <= i < n
Now we must explain what this all means. The main argument here can be deduced from Cauchy’s Theorem which states that for this same group G, where order(G) = (pn)m where n>=1 and where p does not divide m, G contains a subgroup of order p. First off, we see from the initial argument that for (pn)m, p does not divide m. This is a critical argument, since if p did divide m, then the equation could be written as some different (pq) where m is absorbed into the term.
The actual proof of this theorem is quite deep and utilizes the concepts of normalizers of the subgroup H. In summary, this theorem states that if you take the given group G, with the conditions order(G) = (pn)m where n>=1 and where p does not divide m, that for all possible integers i between 1 and the original n, there exists a subgroup pi for the original group G. Additionally, for each subgroup of increasing order, the subgroup with order of exactly one magnitude less forms a normal subgroup of this subgroup. That is to say, for example, that in a hypothetical group G, a group of order p4 would form a normal subgroup of a group of order p5.
Second Sylow Theorem[§]
Let P1 and P2 be Sylow p-subgroups of a finite group G. Then P1 and P2 are conjugate subgroups of G.
In words, this theorem states that all the Sylow p-subgroups of a given group are conjugates to each other. Thus for any two Sylow p-subgroups A and B, A = x-1Bx for some x G.
Third Sylow Theorem[**]
If G is a finite group and p divides order(G), then the number of Sylow p-subgroups is congruent to 1 modulo p and divides order(G).
This theorem is relatively straight forward to explain. We see that given a finite group G such that the order of this group is divisible by p, we can determine the number of Sylow p-subgroups that exist. Thus the exact number of such subgroups is given as 1 modulo p, and thus the number divides order G. This theorem will be better explained through an application, which I will provide now.
Thus now that I have provided the three Theorems of Sylow, I will include some applications of the theorems as to help us understand them better.
Application of the Theorems
Application One[††]:
- There is no simple group of order 84.
Before exploring this example it is necessary to define what exactly is a simple group.
- A group is said to be simple if it is nontrivial and has no proper nontrivial normal subgroups.[‡‡]
Let us proceed now to the application:
Let us take a simple group of order 84.
Through application of the Sylow Theorems, we know that if G is a finite group and pk divides the order of G, then G must contain a subgroup of order pk. We also can determine through manipulation of the theorems that the number of p-subgroups within the given group G is 1+pv with v>=1.
Thus, since 84 can be factorized into 84 = 22 * 3 * 7, there will exist a 7-Sylow (where 7 is p) subgroup. We know if it is not normal, then the group must have 1+7v conjugates where v is greater than or equal to 1 and (1+7v) must divide 22 * 3 * 7. We can see right away that (1+7v) will not divide 7 for any value of v greater than or equal to 1.
Thus we see that 1+7v must equal either 2, 22, 3, 2*3, or 22*3. We notice that all of these possibilities are not equivalent to 1 mod(7), as laid out by the Sylow Theorems, and henceforth we must reject them.
Thus we can say with certainty that the 7-Sylow group is normal and that any group of order 84 cannot be simple.
This concludes our initial example.
Application Two[§§]:
- The Sylow-2 subgroups of S3 have order 2.
This example is a direct application of the Third Sylow Theorem. In this example we will consider the Sylow-2 subgroups of S3, note that the order of S3 equals 3! which equals 2*3. Thus from earlier calculations we know that these subgroups consist of the following:
{po,m1}, {po,m2} and {po,m3}, where po is the identity and m1, m2, and m3 are generators.
Thus we see clearly that the number of subgroups is equivalent to 3. Additionally we see that 3 is equivalent to 1 mod(2), thus the first condition of the Third Theorem is satisfied. Additionally, we can clearly see that the number of subgroups (3) divided the order of the group G (6), and so the number of subgroups divides the order of the group. Therefore, the second condition of the Third Theorem is satisfied.
This concludes the second application which demonstrates the concepts proposed by the Third Sylow Theorem.
Conclusion
In conclusion, we can state that much has progressed since Sylow first published his findings in Mathematische Annalen. His theorems have become the foundation upon which much of contemporary group theory is based. It is curious to ponder whether this quiet high school teacher from rural Norway would have ever imagined the notoriety that he would receive decades after his death for his breakthroughs in the fields of group theory. It can be said with certainty that his theorems will serve as the foundation of many new and innovative concepts for years to come.
Bibliography
- Fraleigh, John. A First Course in Abstract Algebra. 7th. New York: Addison Wesley, 2003.
- Joyner, David. Applications. 12 May 2001. United States Navy. 5 May 2007 <
- "Sylow Biography." 12/96. University of St. Andrews. 5 May 2007 <
[*] All biographical information has been taken from:
[†] All the following definitions have been taken from A First Course in Abstract Algebra.
[‡] A First Course in Abstract Algebra, pg 325
[§] A First Course in Abstract Algebra, pg 325
[**] A First Course in Abstract Algebra, pg 325
[††]
[‡‡] A First Course in Abstract Algebra, pg 149
[§§] A First Course in Abstract Algebra, pg 326