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.
(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.
(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.
This is true by definition of units modulo p (see here).
(a) Let a,b be units modulo p.
(b) ab = bc [Since commutativity of integers with multiplication]
(c) So (ab) ≡ (bc) (mod p)