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Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos.
—Eric Temple Bell

An abstract and crucially important way of representing symmetry and one of the most fundamental concepts in modern algebra. Groups were brought into mathematics in the early 19th century by the radical young French student Évariste Galois as a tool to help solve one of the outstanding problems of his day: to find a formula for solving polynomial equations of order five – quintics – and higher. Galois showed, in notes scribbled down the night before he died in a duel, that no such formula exists. The reason for this is that the possible symmetries, or permutations, of the roots of fifth-degree polynomial equations are more complex than are the symmetries that can be represented by arithmetical formulas. This fact emerged from the development of the idea of a permutation group by Galois and, independently at about the same time, by Niels Abel. Half a century later, another Norwegian, Sophus Lie, showed how important groups are to the whole of mathematics. The theory of what became known as Lie groups links the discrete structure of permutations with the continuous variation of differential equations. Not surprisingly, because group theory forms a common underpinning to algebra and to geometric features such as rotation, reflection, and symmetry, it crops up routinely in modern physics, from the classification of elementary particles to crystallography.

A group is a set whose elements are defined by a single operation. The group is called additive if the symbol for the operation is "+" and is called multiplicative if the symbol "" of multiplication is used instead. But any other symbol can be substituted for these. There is always a unique element (1, for multiplicative, and 0, for additive, groups) that leaves elements unchanged under the defined operation, like a + 0 = a. Also, for every element a there exists a unique inverse b such that, for example, in the case of the additive symbol, a + b = 0 and b + a = 0. Most often, however, the inverse is denoted as a-1. Lastly, the group operation must be associative as in a (b c) = (a b) c. A group is commutative or Abelian if its operation is symmetric, as in a + b = b + a.

Groups come in two types – finite and infinite. The symmetry group of the roots of a polynomial equation is a finite group, because there is only a limited number of permutations possible among the roots of a given polynomial. In contrast, the Lie groups that represent symmetries of solutions of differential equations are infinite because they represent continuous transformations, and continuity carries the potential of an infinite number of changes. Finite groups can be built up from combinations of smaller groups by a process analogous to multiplication. In the same way that a whole number can be written as a product of prime numbers, a finite group can be expressed as a combination of certain factors known as simple groups. Most simple groups belong to one of three families: the cyclic groups, the alternating groups, or the groups of Lie type. Cyclic groups consist of cyclic permutations of a prime number of objects. Alternating groups consist of even permutations – those formed by interchanging the positions of two objects an even number of times. Sixteen subfamilies make up the simple groups of Lie type, each associated with a family of infinite Lie groups. (Confusingly, a Lie group is not a group of Lie type, since the former is infinite and the latter is finite!) Altogether, there are 18 specific families of finite simple groups. There are also 26 simple groups, known as sporadic groups, that are highly irregular and fall outside these families. Five sporadic groups were found in the 19th century by Emile Mathieu. Then came a hiatus until the 1960s, when a suddenly a rush of new sporadics came to light. The most remarkable of these is the so-called Monster group, which appears to be intimately related to the structure of the universe at the subatomic level.

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