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’ can be taken to mean a set of reversible operations satisfying two conditions: one operation followed by another gives a third operation in the same set, and the reverse of each operation is included.
The set of symmetries of an object or pattern forms a group that 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 the rotational 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, and 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.
- There must be an element 1 in G having the property that 1g = g1 = g for any element g in G.
- Each element g in G has an inverse g-1 in G such that gg-1 = g-1g = 1.
- 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.
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.