Friday, February 03, 2006

Groups and Abelian Groups

Group theory is one of the most fundamental and important ideas in all of mathematics. The basic insight came out the theory of algebraic equations (which I will explore in more detail in a future blog)

Evariste Galois first defined the concept of a group in 1832. It took a long time before people generally appreciated the importance of the idea. The mathematical community eventually recognized its importance when Louisville presented a reedited version of Galois's papers in 1846.

At the time Galois and Abel enter the picture, it was well known that there exists an equation for solving the quadratic equation (ax2 + bx + c=0), the cubic equation (ax3 + bx2 + cx + d = 0), and the quartic equation (ax4 + bx3 + cx2 + dx + e=0). It was suspected that no such equation existed for the quintic equation (ax5 + bx4 + cx3 + dx2 + ex + f=0)

Niels Abel was the first to prove that there was no general solution for the quintic equation. His proof is one of the highpoints of mathematical theory and made use of what are today called abelian groups.

Evariste Galois submitted a paper on the solution to the quintic equation as an entry to the Paris Academy's Grand Prize in 1829. The story goes that his entry was returned by Cauchy because it had been predated by Abel's solution which was entered posthumously for the Grand Prize. Galois then generalized his paper to cover all equations quintic or higher. Due to the death of his reviewer, Galois's rewritten paper was never considered and the Grand Prize of 1830 was awarded to Abel and Jacobi.

Galois's proof is one of the most important insights in all of mathematics. He showed that proper understanding of an algebraic equation as a group of permutations determines whether an algebraic equation is solveable through a general algebraic equation. To accomplish this, Galois made use of permutations and what is today known as field theory. I will talk about these other ideas in a future blog. For now, I will focus on groups and abelian groups.

Before defining a group, we need to define four concepts: closure, associativity, identity element, inverse, commutativity.

Definition 1: Closure: When the operation is applied to any two elements of the set, the result is also an element of the set.

This is true of multiplication with integers (an integer * an integer = an integer) and multiplication with fractions ( a fraction * a fraction = a fraction).

It is not true of division with integers (many integer divisions result in a fraction) and it is not true of fractions (consider division by 0 which has an undefined result)

Definition 2: Associativity: if we have three elements, a,b,c, then the (ab)c = a(bc).

This is true of multiplication and addition with integers.

It is not true of subtraction with integers (5 - 6)-1 ≠ 5 - (6-1)

It is also not true of division since (1 / 2) / 2 ≠ 1 / (2/2)

Definition 3: Identity Element: there exists an element such that all elements applied to this element result in the original element.

This is true of addition and multiplication where 0 is the identity element in addition and 1 is the identity element in multiplication.

Definition 4: Inverse: there exists an element that when multiplied with a given element results in the identity element.

This is true of multiplication with nonzero rational numbers. 5 * (1/5) = 1

This is true of multiplication of nonzero real numbers. (1/3) * 3 = 1

This is not true of multiplication with integers. For example, there is no whole number integer x such that 2x = 1.

Definition 5: Commutativity: for any two elements, the operation has the same value regardless of the order, that is, ab=ba.

This is true of multiplication and addition:
1 + 3 = 3 + 1
2 * 3 = 3 * 2

This is not true of subtraction or division.
1 - 3 ≠ 3 - 1
1 / 3 ≠ 3 / 1

Now, we can define a group and an abelian group.

Definition 6: Group: A group is defined as a set of elements and a mathematical operation such that:
(a) Closure: For any two elements of the group, the mathematical operation results in an element of the group.

(b) Associativity: For any three elements of the group, (ab)c =a(bc).

(c) Identity Element: There exists an element i that when combined with any element a, we get: ai=a

(d) Inverse Element: For any element a in the group, there exists an element b also in the group such that ab = i (the inverse element).

Example 1: Addition on integers forms a group.

(a) Closure: integer + integer = integer

(b) Associativity: for all integers (a + b) + c = a + (b + c)

(c) Identity Element: 0: for all integers (a + 0) = a

(d) Inverse Element: For all integers a, a + (-a) = 0

Example 2: Multiplication on integers is not a group.

(a) Closure: integer * integer = integer (true)

(b) Associativity: for all integers (a * b) * c = a * (b * c) ( true)

(c) Identity Element: 1: for all integers a * 1 = a (true)

(d) Inverse Element: 2 does not have an inverse element. (false)

Example 3: Multiplication on nonzero rational numbers is a group.

(a) Closure: rational number * rational number = rational number

(b) Associativity: for all rational numbers (a * b) * c = a * (b * c)

(c) Identity Element: 1: for all rational integers: a * 1 = a

(d) Inverse Element: for any rational number a/b, there exists b/a such that (a/b)(b/a) = 1

Definition 7: Abelian Group: An abelian group is a group that also has the property of being commutative.

Example 1: the addition of integers is an abelian group.

(a) See Example 1 above to see why it is a group.

(b) Commutative: a + b = b + a

Theorem: For a given odd prime p, the multiplication of units modulo p form an abelian group.

NOTE: If you are not familiar with units modulo p, review here.

(1) Closure:

(a) Let a,b be units modulo p.

(b) By definition (see here), there exists a',b' such that (a)(a') = 1 (mod p), (b)(b') = 1 (mod p).

(c) So (a)(a')(b)(b') ≡ 1 * 1 ≡ 1 (mod p)

(d) (a)(b)(a')(b') = (a)(a')(b)(b') [Since integers are commutative]

(e) (ab)(a'b') ≡ 1 (mod p) so by definition (ab) is a unit modulo p.

(2) Associativity:

(a) Let a,b,c be units modulo p.

(b) a(bc) = (ab)c [Since integers are associative]

(c) So a(bc) ≡ (ab)c (mod p).

(3) Identity Element:

We know that 1 is a unit modulo an odd prime p.

(4) Inverse:

This is true by definition of units modulo p (see here).

(5) Commutativity:

(a) Let a,b be units modulo p.

(b) ab = bc [Since commutativity of integers with multiplication]

(c) So (ab) ≡ (bc) (mod p)



Wednesday, February 01, 2006

Èvariste Galois

Èvariste Galois was born on October 25, 1811 in La Reine, France. Galois was educated by his mother up until he was 12. There was some discussion about sending him to college when he was 10 but in the end, it was decided that he should stay at home. In 1815, his father was elected mayor of La Reine.

In 1823, enrolled in school. In 1824-1825, he received good grades, but in 1826, he was forced to repeat a grade because he failed rhetoric. In 1827, Galois enrolled in his first math class.

In 1828, just one year after his first math class, Galois took the entrance examination for the Ecole Polytechnique, the top university. He failed.

Despite the setback, Galois continued his studies of mathematics including work by Lagrange and Legendre. In April 0f 1829, he published a paper on continued fractions which included the proof that reduced quadratic equations are represented by purely periodic continued fractions.

Later that year, a scandal erupted when a vulgar poems were distributed and attributed to Galois's father. The result was more than Galois's father could stand and he committed suicide on July 2, 1829. Just a few weeks later, Galois made his second attempt at entrance to the Ecole Polytechnique. Again, he failed. In December of 1829, Galois entered Ecole Normale.

Galois submitted a paper on the theory of equations to be published. He learned that the same topic had just been covered in a posthumous article written by Niels Henrik Abel. Galois rewrote the article on the conditions whereby an equation is soluble by radicals and resubmitted it. The paper was ver well received and was submitted to Fourier who was secretary of the Paris Academy for the Grand Prize. Unfortunately, Fourier died in April of 1830 and Galois's paper got lost. In June, the Grand Prize of the Paris Academy was awarded to Niels Henrik Abel and Carl Jacobi.

In July of 1830, there was revolution in France. Charles X quickly departed and riots broke out. The head of the Ecole Normale locked the students in a the school to prevent them from joining in the unrest. In 1830, the director of the Ecole Normale wrote an editorial criticizing the students for their behavior. Galois wrote a response defending the students and criticizing the decision to do a student lock up. After writing this reply, Galois was expelled.

Galois next entered the Artillery of the National Guard. In December of 1830, King Louis Philippe disassembled the Artillery of the National Guard because he saw them as a threat to his power. 19 of the guards had been accused of conspiracy but were later released. On May 9, 1831, a great celebration was put together. Galois was there. At one point, Galois raised his glass to make a toast and held up a dagger at the same time. This was taken as a threat against the king. That same evening, Galois was arrested. He was held in prison until June 15 when he was acquitted.

On July 14, Galois was arrested for wearing the uniform of the Artillery of the National Guard which had been outlawed. While in prison, he found out that one of his math papers had been rejected and he attempted suicide. He was stopped by the other prisoners. Finally, on April 29, he was released.

By this time, he was in love with a young woman he had met. On May 30, he entered into a duel. It is believed that it was over the young woman. During the fight, he was severely wounded and died on May 31, 1832 at the age of 20.

Galois's papers were collected and sent out. Eventually, they made their way to Joseph Liouville who was deeply impressed. Liouville presented them to the French Academy in September 1843. These papers were published in 1846 and form the basis of what is today known as Galois Theory.

Today, Galois is considered to be one of the most original and talented mathematicians of all time and Galois Theory is one of the great gems of modern mathematics.


Niels Henrik Abel

Niels Henrik Abel was born on August 5, 1802 in Frindoe, Norway. His father was a vicar in the village of Gjerstad. His father was active politically and was part of the movement that led to Norway's separation from Denmark in 1814.

When Niels was 13 years old, he left home to attend the Cathedral School in Christiania. The school was set up in the new ideals of the time. Instead of corporal punishment, in theory, teachers appealed to a student's sense of honor and decency. Unfortunately, these ideals were not deep in practice. The math teacher at the school would have students copy math problems from the chalkboard and then he would discipline any student who did not learn the lesson. In November, 1817, this math teacher beat a student so badly that the student died. After a protests from the students, the teacher was fired.

It was at this time that a new math teacher came to the school: Bernt Michael Homboe. He had a great love for mathematics that went beyond the standard mathematical education of the day in Norway. He assigned students independent projects. He was very impressed by the abilities of young Niels Abel and helped him to study advanced mathematical topics.

Niels began to greatly excel in mathematics. Despite this, many of the school's teachers were distressed by what was perceived as an overfocus on math. At this time, classical languages were considered the paradigm of a solid education. Abel would regularly visit the public library to study the works of Newton, Euler, Gauss, and Lagrange.

At the time, the University of Christiania offered major only in theology, medicine, or law. There was no track for mathematics. For this reason, Abel studied mathematics on his own. It was very clear that in order to continue his studies, he needed to go abroad. Owing mostly to a lack of funds, Abel stayed at the University of Christiania for 4 years.

In 1823, Abel moved to Denmark where he was able to meet the leading Nordic mathematician of the day, Ferdinand Degen. He lived with his aunt and her husband in Christianshavn. In his private studies, he made progress on number theory and elliptic equations. He also met his future fiancee Christine Kemp.

Abel's first proof was about the impossibility of finding a solution to the quintic equation which he published in 1824 when he was 22 years old. He translated the article to French and he condensed the proof to 6 pages. The resulting article was very difficult to read and was largely ignored by the leading mathematicians.

In 1824, he got a government grant to study mathematics abroad for 2 years. It was during these travels that he met an engineer named Leopold Crelle. In 1826, Crelle launched his soon-to-be-famous math magazine, Crelle's Journal. It was here that Abel would publish most of his work.

It was in Crelle's Journal, that Abel published an expanded version of his proof of the impossibility of solving the general quintic equation, a treatise on the binomial series, and six other submissions.

In July of 1826, Abel headed to Paris. His goal was to impress the very well established Paris Academy. He had worked hard to create a treatise on elliptic integrals. Known today as the Paris Treatise, this is a major mathematical breakthrough. In it, he establishes research areas that are still continuing today and he presented a generality of results that went beyond the mathematics of the day. He submitted this treatise to the Paris Academy in October of 1826. He never heard back. By the end of 1826, he headed back to Norway.

Once back, despite a lack of funds, Abel continued to make great advances in mathematics. He submitted a constant barrage of articles to Crelle's which was not able to publish them as quickly as Abel wrote them up.

In the Autumn of 1828, while Abel was making plans for his marriage, he became very sick and died. He was 26. A few weeks after he died, a letter from Crelle arrived offering Abel an important post in Berlin.

In 1841, the Paris Treatise was published in the Paris Academy's Journal.