# The development of group theory

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The three main areas that were to give rise to group theory are:-

- geometry at the beginning of the 19
^{th}Century,

- number theory at the end of the 18
^{th}Century,

- the theory of algebraic equations at the end of the 18
^{th}Century leading to the study of permutations.

^{th}Century that was to contribute to the rise of the group concept. Geometry had began to lose its 'metric' character with projective and non-euclidean geometries being studied. Also the movement to study geometry in n dimensions led to an abstraction in geometry itself. The difference between metric and incidence geometry comes from the work of Monge, his student Carnot and perhaps most importantly the work of Poncelet. Non-euclidean geometry was studied by Lambert, Gauss, Lobachevsky and János Bolyai among others.

Möbius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group. Steiner in 1832 studied notions of synthetic geometry which were to eventually become part of the study of transformation groups.

(2) In 1761 Euler studied modular arithmetic. In particular he examined the remainders of powers of a number modulo *n*. Although Euler's work is, of course, not stated in group theoretic terms he does provide an example of the decomposition of an abelian group into cosets of a subgroup. He also proves a special case of the order of a subgroup being a divisor of the order of the group.

Gauss in 1801 was to take Euler's work much further and gives a considerable amount of work on modular arithmetic which amounts to a fair amount of theory of abelian groups. He examines orders of elements and proves (although not in this notation) that there is a subgroup for every number dividing the order of a cyclic group. Gauss also examined other abelian groups. He looked at binary quadratic forms

*ax*

^{2}+ 2

*bxy*+

*cy*

^{2}where

*a*,

*b*,

*c*are integers.

*the order of composition of three forms is immaterial*so, in modern language, the associative law holds. In fact Gauss has a finite abelian group and later (in 1869) Schering, who edited Gauss's works, found a basis for this abelian group.

(3) Permutations were first studied by Lagrange in his 1770 paper on the theory of algebraic equations. Lagrange's main object was to find out why cubic and quartic equations could be solved algebraically. In studying the cubic, for example, Lagrange assumes the roots of a given cubic equation are *x*', *x*'' and *x*'''. Then, taking 1, *w*, *w*^{2} as the cube roots of unity, he examines the expression

*R*=

*x*' +

*wx*'' +

*w*

^{2}

*x*'''

*x*',

*x*'',

*x*'''. Although the beginnings of permutation group theory can be seen in this work, Lagrange never composes his permutations so in some sense never discusses groups at all.

The first person to claim that equations of degree 5 could not be solved algebraically was Ruffini. In 1799 he published a work whose purpose was to demonstrate the insolubility of the general quintic equation. Ruffini's work is based on that of Lagrange but Ruffini introduces groups of permutations. These he calls *permutazione* and explicitly uses the closure property (the associative law always holds for permutations). Ruffini divides his permutazione into types, namely *permutazione semplice* which are cyclic groups in modern notation, and *permutazione composta* which are non-cyclic groups. The *permutazione composta* Ruffini divides into three types which in today's notation are intransitive groups, transitive imprimitive groups and transitive primitive groups.

Ruffini's proof of the insolubility of the quintic has some gaps and, disappointed with the lack of reaction to his paper Ruffini published further proofs. In a paper of 1802 he shows that the group of permutations associated with an irreducible equation is transitive taking his understanding well beyond that of Lagrange.

Cauchy played a major role in developing the theory of permutations. His first paper on the subject was in 1815 but at this stage Cauchy is motivated by permutations of roots of equations. However, in 1844, Cauchy published a major work which sets up the theory of permutations as a subject in its own right. He introduces the notation of powers, positive and negative, of permutations (with the power 0 giving the identity permutation), defines the order of a permutation, introduces cycle notation and used the term *système des substitutions conjuguées* for a group. Cauchy calls two permutations *similar* if they have the same cycle structure and proves that this is the same as the permutations being conjugate.

Abel, in 1824, gave the first accepted proof of the insolubility of the quintic, and he used the existing ideas on permutations of roots but little new in the development of group theory.

Galois in 1831 was the first to really understand that the algebraic solution of an equation was related to the structure of a group *le groupe* of permutations related to the equation. By 1832 Galois had discovered that special subgroups (now called normal subgroups) are fundamental. He calls the decomposition of a group into cosets of a subgroup a *proper decomposition* if the right and left coset decompositions coincide. Galois then shows that the non-abelian simple group of smallest order has order 60.

Galois' work was not known until Liouville published Galois' papers in 1846. Liouville saw clearly the connection between Cauchy's theory of permutations and Galois' work. However Liouville failed to grasp that the importance of Galois' work lay in the group concept.

Betti began in 1851 publishing work relating permutation theory and the theory of equations. In fact Betti was the first to prove that Galois' group associated with an equation was in fact a group of permutations in the modern sense. Serret published an important work discussing Galois' work, still without seeing the significance of the group concept.

Jordan, however, in papers of 1865, 1869 and 1870 shows that he realises the significance of groups of permutations. He defines isomorphism of permutation groups and proves the Jordan-Hölder theorem for permutation groups. Hölder was to prove it in the context of abstract groups in 1889.

Klein proposed the *Erlangen Program* in 1872 which was the group theoretic classification of geometry. Groups were certainly becoming centre stage in mathematics.

Perhaps the most remarkable development had come even before Betti's work. It was due to the English mathematician Cayley. As early as 1849 Cayley published a paper linking his ideas on permutations with Cauchy's. In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups. At that time the only known groups were groups of permutations and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. He gives the 'Cayley tables' of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices and quaternions were groups.

Cayley's papers of 1854 were so far ahead of their time that they had little impact. However when Cayley returned to the topic in 1878 with four papers on groups, one of them called *The theory of groups*, the time was right for the abstract group concept to move towards the centre of mathematical investigation. Cayley proved, among many other results, that every finite group can be represented as a group of permutations. Cayley's work prompted Hölder, in 1893, to investigate groups of order

*p*

^{3},

*pq*

^{2},

*pqr*and

*p*

^{4}.

Group theory really came of age with the book by Burnside *Theory of groups of finite order* published in 1897. The two volume algebra book by Heinrich Weber (a student of Dedekind) *Lehrbuch der Algebra* published in 1895 and 1896 became a standard text. These books influenced the next generation of mathematicians to bring group theory into perhaps the most major theory of 20^{th} Century mathematics.

**Article by:** *J J O'Connor* and *E F Robertson*

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