19-12-2012, 06:55 PM
Algebra
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Group Theory
Group theory is a branch of modern algebra.
It could be called the mathematical
language of symmetry.
The first principles of group theory were
discovered by Evariste Galois in the 1830s.
Galois died in a duel at the age of 20.
The exact circumstances of the duel may never be known.
Galois lived during a time of political turmoil in France immediately after the French Revolution. He was critical of the king, expelled from school, joined a militia that was disbanded out of fear it would overthrow the government and was even sent to jail several times for activities considered threatening to the government.
Solvability by Radicals
Galois developed group theory in order to prove the following remarkable theorem:
There is no general formula involving the operations of arithmetic and use of radicals for the solutions of polynomial equations of degree 5 or higher.
Polynomial equations of the form
are equations of degree 2 (because 2 is the highest exponent).
Recall that for equations of degree 2, there is a general formula, the familiar quadratic formula,
involving only the operations of arithmetic (adding, subtracting, multiplying and dividing) and use of square roots.
Applications of Group Theory
Since Galois’ discovery, group theory has grown into a major branch of modern algebra.
As the mathematical language of symmetry, it has many applications in all branches of mathematics and also in computer science, physics and chemistry, just to name a few…
For example, in physics, group theory is used to describe the ways in which elementary particles interact and in chemistry to classify the symmetries of molecules.
A group is a set of elements and a rule to combine them which satisfies certain properties. Sometimes the elements of the group are numbers, but there are many other examples: the elements of a group could be the symmetries of a molecule, the rotations of an object, or the different ways of rearranging a set of elements.
The Rubik’s Cube Group
The Rubik’s cube is another example of a group:
There is a one-to-one correspondence between the distinct
permutations of the cube and the elements of the Rubik’s cube group. Each different permutation represents the result of a single element of the group, just as the different images of the stop sign represented the result of each element in that group.
Different moves of the cube could correspond to the same final permutation and would therefore correspond to the same single element. In other words, a single element of the Rubik’s cube group can be expressed in different ways, using different sequences of moves, just as a rotation of 405º is the same as 45º or the rational number 2/4 is the same as 1/2.
The operation in this group is the combination of moves.
Again, we use the expression “followed by” to represent the operation in this group. In other words, one move followed by another move is itself a different move of the Rubik’s cube.