**Friday 25th July**

**Group Theory**

**Dr Matt Fayers**

http://en.wikipedia.org/wiki/Group_(mathematics)

In mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element satisfying four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation; the addition of any two integers forms another integer. The abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

**Symmetry**

Symmetry (“agreement in dimensions, due proportion, arrangement”) in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, “symmetry” has a more precise definition, that an object is invariant to a transformation, such as reflection but including other transforms too.

http://en.wikipedia.org/wiki/Symmetry

A polygon P such as a square has four axes of symmetry. You can rotate it 90 degrees, 180 degrees and 270 degrees around the centre.

Symmetry of a polygon is a transformation of a plane that appears to leave polygon in the original position. These are things such as reflections and rotations etc.

The square polygon can undergo four reflections at a, b, c and d

Rotation through 90 degrees clockwise = r

Rotation through 180 degrees clockwise = s

Rotation through 270 degrees clockwise = t = 90 degrees anticlockwise

Identity 1 does nothing – it fixes things in place

We can compose symmetries. We can give symmetries x and y. x composed with y. Do x and then y

If a composes with a then the object will return to its original position so

The order is important

http://en.wikipedia.org/wiki/Reflection_symmetry

http://en.wikipedia.org/wiki/Rotational_symmetry

In mathematics, the notion of permutation relates to the act of permuting, or rearranging, members of a set into a particular sequence or order (unlike combinations, which are selections that disregard order). For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters. The study of permutations of finite sets is a topic in the field of combinatorics.

Changing the positions of 1, 2, 3:

Regard permutations as a function

This puts the numbers in a certain order using disjoint cycle notation

So 1 –> 2, 2 –> 3, 3 –> 1

Each bracket is a cycle

http://people.math.sfu.ca/~jtmulhol/math302/notes/4-Permutations-CycleForm.pdf

**Composing permutations**

http://en.wikipedia.org/wiki/Permutation

In mathematics, cycle notation is a useful convention for writing down a permutation in terms of its constituent cycles. This had sometimes been called circular notation and a permutation consisting of a single cycle was called a circular permutation. Modern terminology uses the term cyclic to mean a permutation with one cycle or one set of non-fixed points, and restricts a circular permutation to mean a permutation of objects arranged in a circle up to cyclic permutations, so that there is no fixed starting object on the circle.

http://en.wikipedia.org/wiki/Cycle_notation

Mathematical work with squares stems from the ancient Greeks

http://en.wikipedia.org/wiki/Greek_mathematics

Mathematical work with permutations started in the 18^{th} and 19^{th} centuries

http://en.wikipedia.org/wiki/History_of_mathematics

For both symmetries and permutations ° is a binary operation.

It is a rule to allow you to take two numbers to produce a new number.

A group is a set G with a binary operation ° such that certain numbers are satisfied

http://en.wikipedia.org/wiki/Universal_quantification

http://en.wikipedia.org/wiki/Element_(mathematics)

http://en.wikipedia.org/wiki/Predicate_logic

An associative law

With the 2^{nd} rule there is an element in G such the

Looking back at reflection

Évariste Galois (25 October 1811 – 31 May 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He was the first to use the word “group” as a technical term in mathematics to represent a group of permutations.

http://en.wikipedia.org/wiki/%C3%89variste_Galois

In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.

Originally, Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions.

Further abstraction of Galois theory is achieved by the theory of Galois connections.

http://en.wikipedia.org/wiki/Galois_theory

Galois theory is an associative law that helps to give results for reflections etc.

This is abstract and symbolic algebra

An example of a group is the set of symmetries of a polygon

Groups are ubiquitous in maths but it also appears in physics and chemistry

e.g crystallographic groups

In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals. Informally, two crystals tend to be in the same crystal system if they have similar symmetries, though there are many exceptions to this.

The image below shows that diamond crystal structure belongs to the face-centred cubic lattice, with a repeated 2-atom pattern.

http://en.wikipedia.org/wiki/Crystal_system

Suppose we have a wallpaper with a repeating pattern. This has a group of symmetries

The above example you can translate but you can’t rotate or reflect

In the above example you can translate and reflect but you can’t rotate

In the above example you can translate and rotate about the centre but you can’t reflect.

In the example below you can translate and rotate very carefully

Now for some axes of rotations

In the above example there are horizontal and vertical reflections.

In the above example there are lots of rotations, reflections and translations.

In the above example there are lots of possibilities

Evgraf Stepanovich Fedorov (December 22, 1853–May 21, 1919), was a Russian mathematician, crystallographer, and mineralogist.

His work paved the way for Laue’s discovery and for Bragg’s determination of crystal structures. His most outstanding achievement is the derivation of the 230 symmetry space groups which now serve as the mathematical basis of structural analysis.

http://en.wikipedia.org/wiki/Evgraf_Fedorov

http://www.iucr.org/publ/50yearsofxraydiffraction/full-text/fedorov

There is a well-known proposition called Fedorov’s theorem. It tells us that there are only 17 different types of wallpaper pattern.

A Euclidean graph (a graph embedded in some Euclidean space) is periodic if there exists a basis of that Euclidean space whose corresponding translations induce symmetries of that graph (i.e., application of any such translation to the graph embedded in the Euclidean space leaves the graph unchanged).

Much of the effort in periodic graphs is motivated by applications to natural science and engineering, particularly of three-dimensional crystal nets to crystal engineering, crystal prediction (design), and modelling crystal behaviour.

http://en.wikipedia.org/wiki/Periodic_graph_(geometry)

In 3 dimensions we can think about crystal. What is their group and plane of reflected of symmetries?

In 1879 Leonhard Sohncke listed the 65 space groups (sometimes called Sohncke space groups or chiral space groups) whose elements preserve the orientation. More accurately, he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were really the same. The space groups in 3 dimensions were first enumerated by Fedorov (1891) (whose list had 2 omissions and one duplication), and shortly afterwards were independently enumerated by Schönflies (1891) (whose list had 4 omissions and one duplication). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies.

http://en.wikipedia.org/wiki/Space_group

The aim of group theory is to classify all possible groups.

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

Various physical systems, such as crystals and the hydrogen atom, can be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.

One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

http://en.wikipedia.org/wiki/Group_theory

H and H

This ensures that we can define a quotient group G/H

In mathematics, specifically group theory, a quotient group (or factor group) is a group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

http://en.wikipedia.org/wiki/Quotient_group

The idea is to break down G into two smaller groups

Analogy: factorising integers

G is a simple group if it has no normal subgroups except G, {1}

Analogy: prime numbers

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories.

These groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference with respect to the case of integer factorization is that such “building blocks” do not necessarily determine uniquely a group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated. If the group is finite, then eventually one arrives at uniquely determined simple groups by the Jordan–Hölder theorem. The complete classification of finite simple groups, completed in 2008, is a major milestone in the history of mathematics.

http://en.wikipedia.org/wiki/Simple_group

The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.

http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups

Tests for nonsimplicity

Sylows’ test: Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is equal to 1 modulo p, then there does not exist a simple group of order n.

Proof: If n is a prime-power, then a group of order n has a nontrivial centre and, therefore, is not simple. If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow’s Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple.

Burnside: A non-Abelian finite simple group has order divisible by at least three distinct primes. This follows from Burnside’s p-q theorem.