I.3.3 Group Theory and Symmetry
In modern algebra, groups are systems consisting of a set of elements and a binary operation, that can be applied to two elements of the set, such as multiplication, addition, or any composition. The group must satisfy the following axioms:
1. Closure: the group must be closed under the operation, so that the combination of any two elements produces another element within the group.
2. Associativity: the operation must obey the associative law. For example: .
3. Identity Element: the group must contain an identity element, which when combined with any other element leaves it unchanged. For example: .
4. Inverse Element: that each element in the group has an inverse, so when any element combines with its inverse the result is the identity element. For example: .
If the group also satisfies the commutative law (for example: ), then it is called Abelian or commutative, otherwise it is called non-Abelian or non-commutative. It can be also noticed that both the identity and inverse elements of a group are unique, so there can be only one identity element in a group, and each element in a group has exactly one inverse element. The group is called finite if the number of its elements is finite, and this number is called the order of the group. In the early development of the theory, most of the groups that were considered are realized through numbers, but then more abstract groups were introduced, where the group is usually constructed by some generators and relations.
Gauss developed some parts of group theory, but he did not publish, so Evariste Galois, who coined the term, is generally considered to have been the first to develop the theory. The main mathematical sources of group theory are: number theory, the theory of algebraic equations, and geometry. In 1830s, Galois was the first to employ groups to determine the solvability of polynomial equations.
Group theory provides a powerful method for analyzing abstract and physical systems obeying some kind of symmetry. Group theory has applications in the various fields of science, including physics and chemistry, but groups are so fundamental that they arise in nearly every branch of mathematics and other sciences. Broadly speaking, group theory is the study of symmetry, but it influenced many parts of algebra, because many well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms.
A complete classification of finite simple groups was performed in the second half of the 20th century. From these simple groups, all finite groups can be built. On the other hand, the theory of Lie groups deals with the continuous symmetry, so Lie groups are also differentiable manifolds, with the property that the group operations are compatible with the smooth structure. They provide a natural framework for analyzing the continuous symmetries of differential equations.
Finite groups can be described by writing down the group table consisting of all possible elements, or the group can be defined by its generators and relations, which is called the presentation of a group that is the subject of combinatorial group theory, which studies groups from the perspective of generators and relations.
Symmetry is a mapping of the object onto itself, which preserves its structure. The four axioms of a group are fulfilled by symmetries, because they are closed, since the result of applying two symmetries will still be a symmetry, and the identity is simply keeping the object fixed. Existence of inverses is also guaranteed by undoing the symmetry, while the associativity comes from the fact that symmetries are functions on a space, and composition of functions is associative.
For example, if is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups, which list all possible arrangements of the members of the set. If is a set of points in the plane with any metric space, a symmetry is a bijection of the set to itself, which preserves the distance between each pair of points, called isometry, and the corresponding group is called isometry group of . When angles are preserved, one speaks of conformal maps, which give rise to Kleinian groups, for example. However, symmetries are not restricted to geometrical objects, but include algebraic objects as well.
Groups are important in physics because they describe the symmetries which the laws of physics seem to obey. According to Noether s theorem, every continuous symmetry of a physical system corresponds to a conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to new physical theories. This will be discussed further in section I.1.