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Group Theory
The notion of a group is a vital concept in
modern mathematics, and group theory can be thought of as the
mathematics of symmetry. The term 'group' indicates a group of symmetries, or operations, in which the reverse
of each operation is included, and one operation followed by another gives a
third operation in the same group. The set of symmetries of an object or
pattern always forms a group in this sense, and the group embodies, in an
abstract way, the symmetry of the object or pattern concerned. The application
of groups to serious mathematical problems first arose in the work of Évariste Galois, a young French
mathematician who died after being fatally wounded in a duel at the age of
twenty. Galois studied groups of permutations, where a collection of
objects—in his case solutions to an equation—are permuted among
themselves.
Here is an example in which each operation permutes four people at a bridge table, preserving the two bridge partnerships. Assuming the bridge partnerships are preserved there are eight ways of arranging the seating, shown in the following diagram.
From the arrangement in the top left-hand corner, the other arrangements are obtained by rotating (top row), or by interchanging positions across the dotted lines (bottom row). Notice that the reverse of each operation is included, and that one operation followed by another gives a third operation in the same group. For example a clockwise rotation by 90 degrees takes you from the first position to the second position, and if you follow this by a left/right flip then you reach the last position in the bottom right. The same effect is achieved by a diagonal flip. On the other hand, doing the left/right flip first and the 90 degrees clockwise rotation second, yields the other diagonal flip. The order in which two operations are performed can make a difference to the result.
The way to express this mathematically is to
assign symbols to the different operations. In the diagram above, the symbols
1, x, y, z, p, q, r, s represent the eight operations. The symbol 1 means everything stays in
position. The clockwise rotation by 90 degrees is denoted by x and the left/right flip is denoted
by p. Doing x first, then p yields the diagonal flip denoted by
s, so we write px = s (when composing two operations the
first one is written on the right and the second one to its left). On the other
hand doing p
first, then x
yields the other diagonal flip denoted by r, so we write xp = r. Obviously xp is not equal to px (this multiplication is not
commutative).
The operations in this example were introduced
as permutations that rearrange the four people at a bridge table while
preserving the bridge partnerships, but they can also be thought of as
symmetries of a square, either rotating it or flipping it over. The square
occupies the same space before and after each operation, and this group
comprises all eight symmetries of the square. Another example, given
separately, deals with symmetries of a cube.
The Axiomatic Approach
As mentioned above, the term group of operations, or symmetries, means
that the reverse of each operation is also in the group, and one
operation followed by another gives a third operation in the same group. In
terms of symbols, if G denotes the
group, and if x and y are in G, then so is xy (the operation obtained by first
doing y then
doing x). The
reverse of x is
denoted x‑1,
so if x is in G, so is x‑1. When an
operation is followed by its inverse, the result is that there is no change, so
x‑1x = 1, where 1 denotes the
operation that does nothing.
Mathematicians find it convenient to deal with
groups more abstractly, and not assume that the elements of a group are
operations in any particular sense. They are merely symbols that can be
multiplied together, in a way that satisfies certain axioms. Here is the idea.
Let G be any set endowed with an abstractly defined
product: i.e., if g and h are
two members of G,
then there is defined a third member of G denoted by gh. This product satisfies the following
properties 1, 2, and 3.
1.
There
must be an element 1 in G having the property that 1g = g1 = g for any element g in G.
2.
Each element g in G has an inverse g‑1 in G such that gg‑1 = g‑1g = 1.
3.
For any three elements f, g, and h in G, one has f(gh) = (fg)h.
The third axiom is called the associative law
of multiplication. It is automatically satisfied in a group of operations,
where multiplication gh means operation h is followed by operation g. One can think of f(gh) as meaning first do h followed by g, then do f; while (fg)h means first do h, then do g followed by f. The result is the same in either
case: h followed
by g followed by
f.
Simple groups
Each finite group can be deconstructed into
'atoms of symmetry', called simple groups in technical jargon, though they need
not be simple in the usual sense of the word. The Jordan-Hölder theorem states
that any two deconstructions yield the same collection of simple groups, so the
analogy with atoms is a good one. Finding all finite simple groups (the Classification) led to a table of groups in
several families, along with twenty-six exceptions, the largest of which is the
Monster.
Note that finding all finite simple groups does
not mean one has found all finite groups. It is possible to put two groups
together in more than one way — for example three groups of size 2 can be
put together to form a group of size 8 in five different ways.