A very special case of the theory I'm working on is the following. Consider a "regular" geometric figure - we will take a "tetrahedron" or "pyramid" as a particular example. Suppose we hold the pyramid by one of its vertices (it has 4 of them, 3 on the base and 1 at the top). Suppose in fact that we choose the top vertex, #1, the "tip" of the pyramid. Now rotate the pyramid through one third of a full revolution (that is to say, 120 degrees). If we do this, the pyramid will look the same as before (assuming of course that the sides all have the same colour), since vertex 2 has gone to where vertex 3 was, vertex 3 has gone to where vertex 4 was, and vertex 4 has gone to where vertex 2 was. Such a movement of the pyramid is called a symmetry of the pyramid.
Another symmetry of the pyramid
is given by following the
one described above by another rotation of the same kind; the
two together produce a rotation of two thirds of a full
revolution, that is to say of 240 degrees. This time #2 goes to where #4
was, #3 goes to where #2 was, and #4 goes to where #3 was. Thus there are 2
symmetries associated with each of the vertices, namely a revolution
through 120 degrees, and one through 240 degrees; since there are 4 vertices,
we have found a
total of 8 symmetries.
There are 3 more which are not so easy to visualize.
Consider the edge of the pyramid joining the vertices 1 and 2; we refer
to this as the edge 12. Similarly let 34 be the "opposite" edge
of the pyramid, joining the vertices 3 and 4. Now draw the straight line L
through the midpoints of these two edges.
If we rotate the pyramid
through 180 degrees around L, the edge 12 will be rotated
around its midpoint, through 180 degrees, so the vertices 1 and 2 will be
interchanged. Similarly the vertices 3 and 4 will be interchanged. And the
pyramid is again "taken to itself", giving us another symmetry. (If it's
not clear to you that this rotation does take the pyramid to itself,
consider that fact that the 4 vertices have been permuted by the rotation,
occupying the same points as before, so the pyramid must be in the same
position as before too). There
are 2 other pairs of edges like this, namely 13 and 24, and 14 and 23. So
we get 3 symmetries, one for each of these pairs.
The last symmetry is not a symmetry in the same sense as the others: it is the transformation of the pyramid which is not a transformation! Namely the symmetry consisting of not moving the pyramid at all. This is called the "identity transformation" or simply the "identity", and mathematicians have found that it is very convenient to include it as a symmetry. So altogether we have 12 symmetries of the pyramid.
Now we introduce a way of labelling these symmetries. The (clockwise) rotation through 120 degrees, holding on to the tip 1 of the pyramid, takes 2 to 3, 3 to 4, and 4 to 2. We label it as (234). This symbol is to be interpreted as telling us that each number is taken to the number to the right of itself, except that 4, which has no number to its right, is taken to the number on the other end, namely 2. Any number which does not appear in the symbol, here only 1, is left fixed by the symmetry. The rotation through 240 degrees, with the vertex 1 fixed, is likewise written (243), since 2 is taken to 4 by the symmetry, 4 is taken to 3, and 3 is taken to 2. The other 6 symmetries of this kind, 2 for each of the vertices 2, 3 and 4, are (143), (134), (124), (142), (132), and (123).
The identity symmetry is written ( ), indicating that no vertex is moved since no numbers appear it it. Finally the symmetry which rotates the pyramid around the straight line through the midpoints of the edges 12 and 34 is written (12)(34), consistently with the other symmetries. The other 2 symmetries of this kind are likewise written (13)(24) and (14)(23).
So we have a total of 12 symbols
These symbols exist independently of the pyramid - that is to say we
ignore the fact that they arose as symmetries of the pyramid, and treat
them simply as abstract symbols - and the 12 of them are collectively referred
to as
the alternating group on 4 letters; it is denoted
symbolically by A(4). Since A(4) has a finite number of symbols
in it, namely 12, it is called a finite group. The important thing abouta group is that one can "multiply" two members of the group together. Thus for
example, if we multiply (142) and (13)(24), the answer is the symmetry that
we get by first applying the symmetry (142), followed by application of the symmetry (13)(24). One can look at the pyramid to figure out what the resulting
symmetry is, but in fact it is easier to use the symbols and ignore the fact
that they arose as symmetries of the pyramid. This is done as
follows. The first symmetry (142) takes vertex 1 to vertex 4, and then the
second symmetry (13)(24) takes vertex 4 to vertex 2. Thus the combined effect of following the first symmetry by the second is to take
Now we want to reinterpret a little the way these symbols manifest themselves geometrically. We have viewed them as symmetries of the pyramid. Now imagine the pyramid surrounded by a sphere, with the 4 vertices of the pyramid on the surface of the sphere. When we rotate the pyramid, the sphere gets rotated along with it. So the 12 symbols that make up the alternating group A(4) can also be interpreted as being symmetries of the sphere. Of course the sphere, being a sphere, will have many more symmetries! But for the moment we're just interested in these 12. When we interpret the symbols of A(4) in this fashion, we have what is called an orthogonal representation of the group A(4).
Now A(4) acts as a group of symmetries of the sphere in other
ways as well. For example we can do the following: Suppose we
take some completely arbitrary symmetry of the sphere; let's
call it S. If we "perform S backwards", we get another symmetry T which
"undoes" the effect of S. For example if S is a clockwise
rotation of the sphere through 200 degrees, T would be the counter clockwise
rotation (around the same axis as S) through 200 degrees. Now do the
following: for each symmetry in A(4), first apply S, then apply the symmetry
of A(4), and then T.
So we have defined a new symmetry of the sphere. For example if the
symmetry in A(4) to which we're doing this is (123), the new symmetry
of the sphere (not necessarily of the pyramid, now)
consists of applying S, then (123), then T. This new symmetry
is usually denoted by S(123)T. If we now do this for the other
symbols in A(4), we get 12 new symmetries of the sphere,
namely
Now the crucial question is: are there orthogonal representations of A(4) which are not equivalent to the original one? This is not an easy question but the answer has been known for some time: No, there are no other representations.
The theory of orthogonal representations of finite groups, such
as A(4), are important in other subjects of Mathematics, such
as Topology,
and also in nonmathematical
subjects such as crystallography and molecular theory.