Sylow and his Theorems

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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 Con
tents:


I.

Biography of Sylow

II.

Necessary Definitions

III.

The Three Sylow Theorems

IV.

Applications of the Theorems

V.

Conclusions


Biography of Peter Ludwig Mejdell Sylow


The famous Norwegian mathematician Peter Sylow was born in Christiana (now
Oslo), Norway on Decembe
r 12, 1832
1
. 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 receive
d 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 hi
gh 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, wh
ere 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 t
hat 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
-
18
32.) While teaching both at Christiana
University and at the high school level, Sylow continued his studies and pursued his



1

All biographical information has been taken from:
http://www
-
groups.dcs.st
-
and.ac.uk/~history/Biographies/Sylow.html

own interests. In particular, Sylow initially concerned himself greatly with elliptical
functions, however, after finding particul
arly 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 ((1
809
-
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, S
ylow 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 kn
own 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 to
ok 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 g
roup 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
Sylo
w 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 h
is 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 f
orth 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 f
ollowing definitions, G will refer to a
group and H will refer to a subgroup of G.
2


I.

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 it
self a p
-
group.

II.

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

III.

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.

IV.

Let i
g
: G


G⁷here

i
g
(x) = gxg
-
1

for all x


G⁢e⁴桥⁩ nerm潳琠
慵瑯trph楳i ⁇⁢礠朮†ghu猠瑨攠獵b杲潵p K ⁇⁩猠牥ferred⁴漠慳


conjugate subgroup

of H if K = i
g
[H] for some g







2

All the following definitions have been taken from A First Course in Abstract Algebra.

The Three Sylow Theorems

Now it is time to move on the actual theorems of Sylow:


First Sylow Theorem
3


Let G be a finite group and let order(G) = (p
n
)m where n>=1 and where
p does not divide m. Then
:



G contains a subgroup of order p
i

for each i where 1<= i <= n.



Every subgroup H of G of order p
i

is a normal subgroup of a
subgroup of order p
i+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) = (p
n
)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 (p
n
)m, p does not divide m. This is a critical
arg
ument, since if p did divide m, then the equation could be written as some different
(p
q
) where m is absorbed into the term.






3

A First Course in Abstract Algebra, pg 325

The actual proof of this theorem is quite deep and utilizes the concepts of normalizers of
the subgroup H. In summary, this th
eorem states that if you take the given group G, with
the conditions order(G) = (p
n
)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
p
i

for the original
group G. Additionall
y, 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 p
4

would form a normal
subgroup of a gr
oup of order p
5
.



Second Sylow Theorem
4


Let P
1

and P
2

be Sylow p
-
subgroups of a finite group G. Then P
1

and P
2

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. Thu
s for any two Sylow p
-
subgroups A and B, A = x
-
1
Bx for
some x


G.


Third Sylow Theorem
5





4

A First Course in Abstract Algebra, pg 325

5

A First Course in Abstract Algeb
ra, pg 325

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 strai
ght 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 numbe
r 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
6
:



There is no simple group of order 84.

Before exploring this example it is necessary to define what exactly is a simple group.




6

http://web.usna.navy.mil/~wdj/tonybook/gpthry/node58.html



A group is said to be simple if it is nontrivial and has no proper nontrivial no
rmal
subgroups.
7

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 p
k

divides the order of G, then G must contain a subgroup of order p
k
. We al
so 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 = 2
2

* 3 * 7, there will exist a 7
-
Sylow (where 7
is p) subgroup. We know if it is no
t normal, then the group must have 1+7v conjugates
where v is greater than or equal to 1 and (1+7v) must divide 2
2

* 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, 2
2
, 3, 2*3, or 2
2
*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 g
roup of
order 84 cannot be simple.

This concludes our initial example.


Application Two
8
:



The Sylow
-
2 subgroups of S
3

have order 2.




7

A First Course in Abstract Algebra, pg 149

8

A First Course in Abstract Algebra, pg 326

This example is a direct application of the Third Sylow Theorem. In this example we
will consider the Sylow
-
2 subgroups of
S
3,
note that the order of S
3

equals 3! which
equals 2*3. Thus from earlier calculations we know that these subgroups consist of the
following:

{p
o
,m
1
}, {p
o
,m
2
} and {p
o
,m
3
}, where p
o

is the identity and m
1
, m
2
, and m
3

are generators.

Thus we see clearly t
hat 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
g
roup 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.

Concl
usion

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 t
his
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 fou
ndation of
many new and innovative concepts for years to come.



Bibliography

I.

Fraleigh, John.
A First Course in Abstract Algebra
. 7th. New York: Addison
Wesley, 2003.

II.

Joyner, David.
Applications
. 12 May 2001. United States Navy. 5 May 2007
<http://web.usna
.navy.mil/~wdj/tonybook/gpthry/node58.html>.

III.

"Sylow Biography." 12/96. University of St. Andrews. 5 May 2007
<http://www
-
groups.dcs.st
-
and.ac.uk/~history/Biographies/Sylow.html>.