Introduction

Syllabus: In this section we give an intuitive and non - technical introduction into group theory as a tool to analyze symmetries and also outline directions of the later theoretical development.

Groups measure symmetry

Group theory is the mathematical language for analyzing symmetries and making them amenable to measurement and computation. The following picture shows three objects. One of them is not symmetric at all, the second one has one symmetry, which can be described by a reflection in a line. The third one is `more symmetric' by allowing three reflections in three different lines:

Here a `symmetry' or `symmetry operation' is a distance preserving transformation in the plane, which may move individual points but maps the given object onto itselft as a whole. A first, not very precise definition of `symmetry groups' is the following: The symmetry group ${\rm Sym}_A$ of the object A is the set of those transformations that preserve the "structure" of A.

Notice that the reflection in y is in ${\rm Sym}_\Delta$, whereas the reflection in x is not. A very important observation is the fact, that in some respect symmetry operations behave like numbers:

- they can be multiplied (=composed): $a\circ b :=$ `first apply b to A, then apply a ';

- they satisfy the associative law $a\circ (b\circ c) = (a\circ b) \circ c.$

But, as the following example shows, unlike numbers, symmetry operations in general do not commute, i.e. $a\circ b \ne b\circ a.$

Let us write down the full symmetry group ${\rm Sym}_\Delta$ of the regular triangle:

This picture shows that ${\rm Sym}_\Delta$ contains the `identical transformation' ${\rm id}$, a clockwise rotation r of 120^o around the centre, with ${\rm id}\ne r^2:=r\circ r$, $r^3:={\rm id}$ and the reflections s_a in a, s_b in b and s_c in c.

We saw that the smallest positive exponent $m\in \mathbb{N}$ with r^m=1 is three; one says the element r has order three. Similarly the reflections have order two, whereas the identity ${\rm id}$ is the only element of order 1. Every symmetry operation has to permute the three corners of $\Delta$ in some way; on the other hand all $3\cdot 2=6$ permutations of the corners can be realized by these symmetries. This tells us that ${\rm Sym}_\Delta$ coincides with the set $\{r,r^2,r^3={\rm id},s_a,s_b,s_c\}$. One also says ${\rm Sym}_\Delta$ is a finite group of order 6.

Notice that each of these symmetries can be described uniquely by the permutation, which it induces on the three corners. For $a,b,c\in \{1,2,3\}$ let (a,b) denote the permutation that swaps corners a and b and leaves c fixed, while (a,b,c) maps a to b, b to c and c to a (`cycle notation'). Furthermore let () denote the identical permutation. Then r corresponds to (1,2,3), s<sub>a</sub> to (1,2), s<sub>b</sub> to (2,3) and s<sub>c</sub> to (1,3). The set $\Sigma_3$ of all bijective functions from the set $\{1,2,3\}$ to itself can be viewed as the `symmetry group' of that set. Even though these bijections are mathematical objects which are quite different from symmetries in the plane, we see that the groups ${\rm Sym}_\Delta$ and $\Sigma_3$ are very similar. In fact all their computational properties are the same and they cannot be distinguished algebraically. Therefore one says ${\rm Sym}_\Delta$ and $\Sigma_3$ are isomorphic groups and writes ${\rm Sym}_\Delta \cong \Sigma_3$. A more precise definition will be given later. Obviously it is a major task of group theory to identify, whether or not two groups are isomorphic and also to get more information on the possible `isomorphism types' of groups.

Abstract groups

The general notion of an abstract group summarizes the essential similarities of all symmetry groups

A group G is

(i) a set with multiplication of elements $a,b \mapsto a\circ b;$

(ii) with neutral element e such that $a\circ e = e\circ a = a$;

(iii) satisfying the associativity rule $a\circ (b\circ c) = (a\circ b)\circ c$

and (iv) allowing unique solutions of equations $a\circ X = b\ \Rightarrow \ X = a^{-1}\circ b.$

Example: The integers $0,\ \pm 1,\ \pm 2,\ \pm 3,\cdots$ form a group under addition $n,m \mapsto n+m$ with neutral element 0.

The natural numbers $1,\ 2,\ 3,\ \cdots$ do not form an additive group for two reasons: there is no neutral element and there is no solution to equations like 3+X=1.

Decomposition

Not only the elements of groups, but groups themselves `behave like numbers'. As it is known from elementary arithmetic, large natural numbers can be decomposed into a product of smaller ones, with the prime numbers being the `atoms' in this process:

$$\ 12=2^2\cdot 3\ \ \ {\rm or}\ \ \ 6100875 = 3^2\cdot 5^3\cdot 11\cdot 17\cdot 29.$$

In a similar way one can decompose groups:

notice that every element of ${\rm Sym}_\Delta$ is either a rotation r^m, or a rotation $\times\ s_a$, hence

$${\rm Sym}_\Delta = R\cdot S = \{ x\circ y\ \vert\ x\in R,\ y\in S\}$$

with the subgroup of rotations R and the subgroup $S:= \{{\rm id}, s_a\}.$ This of course makes more sense in the analysis of really LARGE groups, like the symmetry group RC of Rubik's Cube:

Here RC is the group of all symmetry operations, generated by composing the six 90^o - rotations around the central squares. It turns out that RC is a group of order

$$43252003274489856000 \sim 4.3 \times 10^{19},$$

which is the total number of possible configurations of the cube. A group theoretical analysis reveals that

$$RC\cong [\ (\ {\rm Alt}_3\wr \Sigma_8\ )\ /\ Z\ \times\ (\ \Sigma_2\wr \Sigma_{12}\ )\ ]': \Sigma_2,$$

where `$\cong$' means `isomorphic to' and the symbols ${\rm Alt}_3, \Sigma_8, \cdots$ denote well known types of smaller groups, and the symbols $\ \wr\$, $\ \times \$, $\ /\$, $\ :\$ $\cdots$ denote certain product, quotient and amalgam constructions in group theory. In the group theoretical process of `decomposition', the atoms are the so called

Simple groups

Not all subgroups of a given group G are are equally well suited to decompose G. The useful ones are called normal subgroups or invariant subgroups. A simple group S is a group whose only normal subgroups are $\{{\rm id}\}\ \ {\rm and}\ \ S\ {\rm itself}.$

An (admittedly very rough) comparison with numbers looks as follows:

Group G $\longleftrightarrow$ natural number $n\in \mathbb{N}$

ordinary subgroup $H\le G$ $\longleftrightarrow$ smaller number $m\le n$

normal subgroup $N\triangleleft G$ $\longleftrightarrow$ divisor $d\ \vert\ n$

simple group S $\longleftrightarrow$ prime number.

Like prime numbers in arithmetics or like atoms in chemistry, a simple group S cannot be decomposed group theoretically.

Simple groups are structurally simple

but can be technically complicated !

Simple commutative groups

In the special case of commutative groups, the analogy between groups and numbers is much closer than in the general case. It turns out that for each prime number p there is exactly one isomorphism type of a simple commutative group which can be described as follows:

Consider the regular p - gon:

and let C_p denote the symmetry subgroup of rotations.

Try to prove ( = convince yourself) that: there are no subgroups besides $\{{\rm id}\}$ and C_p.

It turns out that the group C_p is a finite simple group and that all commutative finite simple groups are of this form. Of course this cannot be proved here, as we have not yet given a precise definition of simple groups. If we include non - commutative groups, the situation becomes much more complicated and therefore much more interesting. The smallest non commutative simple group has order 60 and is isomorphic to the alternating group ${\bf Alt}_5$ consisting of all even permutations   of 1,2,3,4,5. These are the one - to - one mappings of $1,\cdots,5$ with an even number of misplacements  :

$\left( \begin{array}{ccccc} 1&2&3&4&5\\ 3&4&1&2&5\\ \end{array} \right)$ is even with the 4 misplacements (3,1), (3,2), (4,1), (4,2).

$\left( \begin{array}{ccccc} 1&2&3&4&5\\ 3&2&1&4&5\\ \end{array} \right)$ is odd with the 3 misplacements (3,2), (3,1), (2,1).

$\left( \begin{array}{ccccc} 1&2&3&4&5\\ 3&4&1&3&5\\ \end{array} \right)$ is not a permutation, because it is not one-to-one.

The simplicity of the group ${\bf Alt}_5$ is the deep reason, responsible for the fact that equations like

X⁵+aX+ b =0

or of higher degrees cannot be solved by finite additions, multiplications, and root extractions. This was first shown by Evariste Galois (1811-1832) who is the founder of group theory as a serious mathematical discipline.

Classification of finite simple groups

Let us further develop the analogy between arithmetics and group theory. Recall Euclid's theorem

Theorem [Euclid] (400 b.C.) There are infinitely many prime numbers.

But, to the benefit of all cryptologists, there is no known general rule to construct them all!

The situation of (finite) group theory is quite different. A major result of last century's mathematics was the following

Theorem(1981) (Many many authors) All finite simple groups are known!

It turns out that there are

-

Infinite series of uniformly constructed groups; e.g. C_p, ${\rm Alt}_n$ for $n\ge 5$;

-

26 `sporadic' simple groups with no uniform pattern.

• All non - commutative finite simple groups have even order!
• The proof is more than 5000 pages long, scattered over more than 500 research publications.

Sporadic simple groups

These are still sort of mysterious objects; they do not fit into any of the infinite series and even though they are known explicitly there is no unified rule to construct them. The smallest ones were disccovered by E. Mathieu (1861) in the context of the following game:

Given 11 persons, set up 66 committees containing 5 persons each such that any 4 persons belong to exactly one committee! The set of all arrangements has the simple symmetry group M₁₁ of size 7920, which is also the total number of these arrangements. In a similar way Mathieu discovered M₁₂, M₂₂, M₂₃, M₂₄ which were the only known sporadic simple groups for a century. In 1966 Z. Janko discovered the next sporadic simple group, which is called the first Janko group J₁ and has order 175560 = 23.3.5.7.11.19.

More information on finite simple group can be found in Online - ATLAS of finite simple groups under

http://www.mat.bham.ac.uk/atlas/html/

The Monster and its Baby

The latest and largest sporadic group was first conjectured and then constructed by B.Fischer and R.L.Griess in the late 1970's.

It is called the `Monster - group' $\mathbb{M}$ and is of size

$$\vert\mathbb{M} \vert = 808017424794512875886459904961710757005754368000000000 =$$

$$2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13^3\cdot... ...19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71\sim$$

$$8\times 10^{53} \ge 1000\times \vert\{\rm Atoms\ in\ the \ Earth\}\vert .$$

To store one single element on a computer one needs a matrix of format $196882 \times 196882$ with 0 - 1 - entries (5 Gigabytes)! One multiplication inside $\mathbb{M}$ takes about 45 hours on the currently fastest computers!

The monster contains 20 of the 26 sporadic groups as `substructures'. One of the most interesting is the Baby-Monster $\mathbb{BM}$ of size

$$\vert\mathbb{BM} \vert = 4154781481226426191177580544000000 =$$

$$2^{41}\cdot 3^{13}\cdot 5^6\cdot 7^2\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 31\cdot 47.$$

The group $\mathbb{M}$ and $\mathbb{BM}$ contain important structural information in field theory of modern physics ("monstrous moonshine").

The way forward

In arithmetic the construction of large numbers from smaller prime numbers is just multiplication and therefore `known'. In group theory, the process of constructing larger groups from simple ones is much more complicated and therefore still a very active field of research. Therefore the classification of finite simple groups is not the end but the beginning of the story about finite groups.

Things which remain to be done are among others:

to `streamline' and revise the proof of the classification theorem;

to analyse the explicit structure of the finite simple groups (with the help of modern computers);

to investigate how to construct all composite groups from the simple ones.

The following can be viewed as a general motto:

Now we know the bricks, so we have to learn

how to build the house!