#6: Galois Lays the Foundations for Group Theory

Mathematician Évariste Galois lived a short life: 1811 to 1832 when he was killed in a duel, most likely by a romantic rival. However, during only the last 4 of those 21 years, Galois explored much of what is now considered the beginning of group theory.  He officially published almost nothing during his lifetime, but his journals and work were gathered and published postmortem in 1846.  Galois was even the first to use the word “group” to describe a set (see moment #4 for a definition of sets) with an operation, (eg. addition or division) that composes some of the elements in the set to get other elements. Additionally, the operation in consideration with the set must obey the following rules in order make the set a “group under the operation” (let “&” represent the symbol for the operation, more detailed explanations and examples are included in the exercises for this page.)

1) Closure: for any two elements in the set, x, and y, if x&y=z, z must also be an element in the set.
2) Identity: The set must contain an element i, the identity, such that for all elements, x, in the set, x&i = i&x = x
3) Inverses: For every element in the set, x, there must exist an element y such that x&y = y&x = i. When this is the case, y is called the “inverse” of x.
4) Associativity: If a, b, and c are elements of the set, then (a&b)&c = a&(b&c)

When these four properties are true for a set under a specified operation, the set is a “group under the operation.” There is an additional fifth property that is true for some, but not all groups:

Commutativity: If a and b are elements of the set, then a&b=b&a. If a group commutes, it is called “abelian.” Thus, the group theory joke: What's purple and commutes? An abelian grape! (BWAHAHAHAHAHAHAHA!)

Why are groups useful? Because they appear everywhere! Operations more interesting than the standard arithmetic ones include things like permutation of lists, valid movements of a chess-board knight, and rotation of shapes. For example, the rotations and reflections of a cube make a group! To “compose” two elements, preform one operation, then the other. For example, if you rotate the cube 90 degrees clockwise, and then reflect it across the plane defined by the axis of that rotation and the vertical axis, you get the same final position as if you had reflected it across the plane across both of the the top right and bottom left corners. Note that this symmetry group is non-abelian: if you reflect and then rotate, you don't get the same result as rotating and then reflecting. Groups are often applied to study the symmetry of molecules and particles.  The positions and moves of a Rubix cube are also a group.

Galois jumped into number theory using group theory as a tool, and began work
on the mathematics now known as Galois theory. Galois theory is an advanced application of group theory that can be used to, among other applications, elegantly explain why it's not possible to trisect angles using "compass and straight edge" techniques, why it's only possible to algorithmically draw some regular polygons, and why there is no algebraic formula for the roots of general high-degree polynomials. Especially given that his mathematical 'career' was less than 5 years long, Galois made a huge contribution to mathematics, opening a field that still contains many beautiful unsolved problems!

Exercises:

1) When is a set a group? (Examples and exercises to go with the four group-requirements defined above)
2) Symmetry Groups of Polygons
3) Cyclic Groups and Generators
4) The Rubix Cube as a Group
5) Open problems in Group Theory
6) Galois Theory

References:
Galois Biography
Mathworld: Group
Symmetries of a Cube
Applications of Group Theory
Symmetries in Chemistry
Wikipedia on Galois Theory
Wikipedia on Evariste Galois
Cayley Graph of Cube Rotations

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