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A History of GroupTheoryWilliam Komp
History of Mathematics
Dr. Davitt


The origins of group and field theories, modern algebra, lie in developments of algebra and geometry. Much of the work of Euclid and the Islamic mathematicians are directly related to the development of modern algebra. It is the apparent differences between the "numerical" symbols and the quantitative geometric lengths that these two distinctive approaches made that ultimately came together and formed the basis of group theory. The advances made in non-Euclidean geometry and the works of Gauss, Euler, Lagrange,etc. on the solvability of arithmetical equations of order five or higher help to lay the foundations for Galois and Abel to develop and release their insights. From Galois and Abel the next generation of mathematicians were able to pick up and expand the fledgling group theory ideas, Dedekind,Kronecker, Camille Jordan, and others (et al). These mathematicians developed the notions and used group theory, as we know it today. This paper will focus on preeminently on the development of group theory from the question of solvability of equations, and geometry.
This raises the question of traits do all groups have in common, and as an extension on to this what additional traits do all fields share. There have been many interpretations of the structure of groups. H. Weber's notion of a group has for axioms, but it turns out that one of his axioms is unnecessary when compared to the modern definition. The modern definition of a group is as follows:
Given any set A, we say that A forms a group under the binary operation @ if and only if it satisfies the following four criteria.
I) Given any two elements in A, then their product under the binary operation @ will be in A [closure property]
II) Given three elements in A, these three elements will satisfy the associative property.
III) In A, there exists an element such that the product of this element with any other element in A, namely ©,will generate the element © back again.
IV) There exists an element ©' in A such that the product of © and ©' will give the element mentioned in III.(Gallian)
Group Theory has enormous potential in practical applications in areas other than that of mathematics. In mathematics, group theory is the basis of real analysis. Namely, the set of R, the real numbers, must be complete and satisfy all of the properties of an ordered field (Sherwood 14-19). This provides the necessary algebra to begin proving and developing axiomatically all of Calculus. The analogy can also be made with the complex numbers, with some increased complexity in the area of complex analysis. The properties of a field are very similar to those of a group.
Group theory can also be used to develop the algebraic structures of linear spaces and vector spaces. It was Hermann Grassman was a major influence of Guiseppe Peano. Peano, who was the first mathematician to give an abstract definition to a vector space. He developed abstract vector spaces to supply a geometrical calculus. In developing this system, Peano came up with several criteria, a few of which satisfy the necessary conditions of group theoritic structure.
In the field of physics, Group Theory plays many prominent roles. It is used as a fundamental basis of the space of Quantum mechanics, this in particular is the work of Dirac who used Hilberts work as a starting point. In vector spaces, an advance use of geometry has helped to lead the way into their development. Whenever geometry is used, such as used in vectors, symmetries often arise and these symmetries can be analyzed using group theory. By applying vector spaces to geometry, Hilbert added to the algebraic structures of vector spaces. In classical mechanics, geometric symmetries developed by Lagrange and Hamilton can also be explained by group theory.
These geometric symmetries, noted by Lagrange in classical mechanics, assisted in the advancement of geometry in the 19th century and in particular the development of non-euclidean geometry. Non-euclidean geometry represents a break with the traditional ideals of the time, i.e. the ancient greeks. From humble beginnings in the theories of Euclidean geometry began to evolve slowly into a "non- euclildean" form.
The first modern evolution in geometry occured in the very late 17th century with the work Saccheri (1667- 1733), in 1697. He brought up the first and probably one of the most important questions of Euclidean Geometry, the fifth postulate. The dispute over Euclids fifth or parallel postulate was centuries old. The question that was proposed over this postulate was wether or not this postulate was as a theorem or was it a true postulate. Saccheri believed that the fifth postulate was really a theorem and could be proven by the previous four postulates spelled out in The Elements. Saccheri looked at two lines that are incident upon a third line such that the two lines form a right angle with respect to the third. Saccheri thought that by using various properties of angles enclosed by similar triangles, he could prove that if one completes the rectangle the angles formed by the completion must be equal and add up to 2 right angles. He ultimately failed to show that the angles must sum to 2 right angles.
Saccheris work was picked up later in the 18th century by Johann Lambert, (1728-1777),he was unable to expand beyond Saccheris work. Lambert began by using a similar construct as that of Saccheri; Lambert considered a quadrilateral and three of the interior angles of this quadrilateral. He presumed that three angles of the quadrilateral were right, then the fourth would either be right, acute, or obtuse. He was able to rule out accute, but right or obtuse he was unable to resolve. He ultimately did not publish his results. However, Lambert showed the limitation of Euclidean Geometry.
The main interest in geometry in the 18th century was the relation of geometry to analysis. Alexis Clairaut (1713-1765) considered curves as intersections of other surfaces called space curves by Katz; we know these curves in a orthogonal cartesian coordinates or the "x-y-z plane". He used differential calculus to derive the equation of a sphere, a paraboloid and curves generated by a rotation about the z-axis. Notice that a rotation about the z-axis makes any such curve symmetric about any rotation of the z-axis, and he noticed that the cartesian axis could be rotated this rotation had no effect on any given sphere. Later, both of these symmetries were shown to form cyclic groups and also can be represented as permutations. Leonard Euler (1707-1783) came at space curves through a system of parametric equations or equations that could be used to describe a curve in the cartesian coordinate system. He then arrived at the direction cosines. Direction cosines are angles, usually denoted by ?????, that any point or curve in cartesian 3-space will form angles with respect to the axes. It is interesting to note that there is a direct relationship between direction cosines and the usual 3-vector space used in elementary physics. This precursor to the modern geometric vector system used in physics and mathematics, which would later be defined at the end of 19th century as having group structure.
The major influences in the evolution of the group theoretic concept in geometry came out off our major principles, according to Wussing:I) The unity of geometry and metric and a rise in the distinction between projective and metric geometries.
II) The development of non-Cartesian co-ordinates.
III) The full realization of Lamberts work in non-Euclidean geometries.
IV) The full abstraction of an nth dimensional geometry.The works in space curves and Lamberts Non-Euclidean Geometry was the driving influence in the works of J.V. Poncelet, (1788- 1867). He was able to make the distinction between projective geometry and non-projective geometry. Poncelet introduced the abstract properties of central projections and the properties that are lost when this projection is made. He predicted further developments in geometry that would offer classifications of geometric objects. Ultimately, these classifications took the form of the works of Felix Klein (1849-1925) and his Erlangen Program in1872.
The work of August Ferdinand Mobius (1790-1868), a student of Gauss, was strongly influential in the extension of the traditional coordinate systems. He looked at what we call "barycentric" coordinates. He used fundamental points, what he thought of as centers of gravity, to define his geometry. He could then find a transform from "ordinary" (A. F. Mobius) to "barycentric" (A.F. Mobius) coordinates. These coordinates are said to be "homogeneous". This is the main influence in the development of the theory of vector spaces. Mobius, like Poncelet and completely independent of him, anticipated Kleins Erlangen Program; namely the distinction could not be made in Mobius work from similar geometries and congruent geometries. Mobius looked at the descriptions of elementary geometry and the relationships of these figures. It is Kleins Erlangen Program that ultimately linked geometry and abstract algebra.
Julius Pluckers (1801- 1868)work on "triangular" and "tetrahedral" (Plucker) coordinate systems isa practical application of Mobius "barycentric" coordinates. Plucker published the results of his work the same year that Mobius published his work. Plucker, upon reading Mobius work, deferred and suggested that Mobius work should be the primary area of study. The coordinate systems used by Plucker would ultimately be examples of group theorys uses in geometry.
Steiners work with synthetic geometry, although not as fruitful in the end as Mobius work, did lead the way for later mathematicians to apply the theory of forms from Euler and Gauss in geometry. Steiner attempted to generate all of the properties of a geometry from the use of its "fundamental forms" (Steiner). However, he found that using these forms did not allow him to approach imaginary geometric forms. This work did point out the need for the use of number theory and in particular the theory of forms from number theory and algebra if this approach to organizing geometry were to work.
Number theorists and algebraists prior to the publishing of Galois work mainly were involved with properties of real and complex numbers and solving equations respectively. Katz points out that in the 16th century, Gerolamo Cardano (1501-1576)and Lodovico Ferrari (1522-1565) along with Niccolo Tartaglia (1499-1557)came up with a way to solve a generalized cubic equation. Cardanoand Ferrari later went on to solve fourth degree equations by radicals. Following their works in the 18th century, Edward Waring worked on solving the cyclotomic equation, i.e.- x^n 1 =0. He pointed out for later generations, that this equation could be solved using products of roots. After Waring, Alexandre-Theophile Vandermode and Joseph Louis Lagrange independently developed more mathematical machinery, particularly "Lagrange Resolvents". Vandernode was able to apply this to solving a cyclotomic equation of degree 11. He used primitive roots in order to break the 11th degree equation down to a quintic equation. He used trial and error to find the roots of the fifth degree equation. He noted that a proof would be required in general to solve the nth degree case.
Lagrange started off considering a general cubic equation and known quartic equations. He took a general nth degree equation and and broke it down into reduced equations. Lagrange realized, in modern terms, that the permutations of roots could give some valuable assistance when trying to solve a particular equation. However, Lagrange never developed neither did he study the properties of these permutations, i.e. he never developed a "calculus" for the permutations (Wussing 79). From Novy, the work that Lagrange did with reduced equations and permutation of roots failed to assist him in the answer to the question as to whether the fifth degree equation was solvable or not.
Following Lagrange, Gauss and Ruffini tried their hands at answering the question of solvability of equations. Ruffini, a student of Lagrange and similar to him in mathematical temperment, worked on the question of solvability of a general nth degree equation. Ruffini published several papers claiming that he had solved the question of solvability of a 5th degree equation. Gauss work was considerably different than Ruffinis. Gauss did not try to answer the question of the fifth degree equation. He did work on the solution of the generalized nth degree cyclotonic equation. He developed a proof to the solvability of this equation, and in so doing he developed a generalized method for generated cyclic groups. Gauss became aware of the isomorphisms between additive groups and multiplicative groups while working on the generalized cyclotonic equation. Gauss clearly stated and proved the Fundamental Theorem of Algebra, which states
In the notation of Gauss, every algebraic equation of degree m can be written as x^m+A*x^m-1 +B*x^m-2++M=0 or X=0. The so-called "Fundamental Theorem of Algebra" says that every polynomial X with real or complex coefficients can be factored into linear factors in the field of complex numbers. [Waerden 94]
Gauss thought so highly of this theorem that he gave four different proofs of this.
After Gauss and Ruffini, there was a change in the approach to algebra. Cauchy, Abel, Galois, etc came onto the scene. Cauchy was the first to employ a permutation in the same way that we use it today. He called them "substitutions" (Katx 599). He also used "group of permutations",keeping with the notation of today. Abel was able finally able to show that a fifth degree equation was insolvable by addition, subtraction, multiplication, division, and roots. It is a note of irony that he was looking for and thought he had obtained a solution for the generalized fifth degree equation. He was then easily able to expand this notion to include any equation of higher degree. Abel also contributed to the formation of group theory, and in particular those groups which now bear his name, i.e. Abelian Groups. He stated two months before he died: Given an equation which are solvable by roots, and if the roots can be expressed as a rational function, l then two roots x*l and x2*l then x*x2*l=x2*x*l This is the commutative property. The uses of this are profound in group theory.
This brings the story up to Galois. He was born on October of 1811. He died in a duel on May 30,1832. He was an unconventional mathematician to say the least. His proofs and propositions were "vague" and "incomprehensible". Despite this discrepancy, he was the mathematician that took the great step into group and field theory. In 1830, Galois proposed a proposition1 (Katz 601 and Waerden 107). It says, to quote Waerden, "There isa group of permutations of the letters a,b,c, such that
1.) Every Function of the roots, invariable under the substitutions of the groups, is rationally known;
2.) Conversely, every function of the roots rationally known is invariable under the group."
He developed from this Proposition the substitutions of the groups into the idea of a group of the equation. To develop his notion of a field, Galois expands on the congruence calculus of Gauss. Galois proposes what the decompositions of a group are using subgroups H, which are normal to the group. By normal, it is meant that the left and right cosets are equivalent, i.e.- {Ht}={tH}. Using this he says that an equation is solvable rationally, if one can find normal subgroups until the indices of the subgroups become prime numbers (Katz603). It was not until several years after Galois duel that his work was widely circulated. Liouville, in 1846, published every singlepiece of Galois theory he could find in his Journal de mathematiques pureset appliquees.
There were still some steps missing for Cayley to step on to the scene. Enrico Betti (1823-1892)showed that related Galois group to a group of permutations. J.A. Serret (1819-1885) linked Cauchys work with Galois groups.
Camille Jordan (1838-1922)compiled Serrets work. He showed that group theory had evolved into an explicit mathematical construct. He defined a group "to be a system of permutations of a finite set with the condition that the product (composition) of any two permutations of the system" is also a permutation (Katz 603). Jordan defines transforms from one element of a finite permutation (A) to another element in another finite permutation (B). If it happens that the set A and B coincide then Jordan says that B is permutable to A. Jordan also gives form to Galois work. He says that an equation is solvable in terms of radicals if and only if it is in a solvable group. Using this and the interchanging of groups of permutations, i.e. Abelian equation, is solvable by radicals. This means that every Abelian Group are completely decomposable to cyclic groups of prime power order. Jordan worked mainly with permutations, and if a way could be found to relate permutations to other parts of algebra then Jordans work would hold in more general ways. It took Arthur Cayley to relate permutation groups to other groups.
In 1854, Arthur Cayley(1821-1895) proposed an addition to Galois theory of groups. He was able to state the theorem that now bears his name. "Every Group is isomorphic to a group of permutations. (Gallian 106)." He was the first mathematician to realize that various groups could be identical to groups that apparently are not similar, that is a group whose elements that are entirely unrelated could be soon to be identical under a given binary operation. Also Cayley developed the "Cayley" table, which is a table that generates the entire finite group by performing the binary operation of one element with another in matrix or pseudo-matrix form. The development of the Cayley table requires a general understanding of the form of groups themselves. Cayley defined a group as a set of symbols, 1, a, b, where each of the elements, in the set, are unique. Cayleys work was limited to groups with a finite number of elements. Along with the set of the group, there is a binary operation such that when one element operates on another then the result is still an element in the set.
Independent of Cayley, Leopold Kronecker (1823-1891) was perhaps the first mathematician to link Neils Henrik Abels notion of commutativity into the set of a finite group,i.e.-abstract abelian group. Given a finite group (G,*)-where G isthe set of the group and the * is the binary operation; and give a and b being elements of the set G; then a*b=b*a. Kronecker then proceeds to propose and prove the fundamental theorem of finite Abelian groups.
It was Felix Klein (1849- 1925)in his famous Erlangen Program who made the final link from the emergin ggroup theory and our present group theory. He sought to use higher dimensions in his work in algebra and geometry. Using Gauss, Cayley and others, he realized in his work was one could define a set of transforms that leave invariant a certain amount of characteristics. Klein was able to link permutations to geometries, thus establishing the link of group theory with geometry.
This left group theory in an unrefined state. There was much that had to be polished. There was uncertainty in what were the necessary properties for groups. The area of infinite groups was also unexplored. Starting with a permutational aspect of group theory, it is very hard to imagine an infinite permutation. An example of an infinite permutation would be to take every integer a and map it into a+1. The work of Galois, although using field theory, was also very mysterious even after the formal development of abstract algebra. It was the work of many of the mathematicians from Klein until the present, e.g. Walther von Dyck (1856-1934), Heinrich Weber (1842-1913), Sophus Lie (1842-1899), et al. Walther von Dyck developed the concept of what is today called a free group. When a free group is given a number of operations then the most general group that can be built from them using the powers of these elements and the inverses of the powers. Dyck also developed the concept of an "isomorphism" between two groups. The isomorphism mean essentially two groups are identical. All of Dycks work satisfies the properties that are required by the axioms of group theory, as they are known today.
Heinrich Weber starts with the notion of the order of a group. Weber said that a group with h elements is a group of order h, where h is finite, if and only if:
I) If two elements of the groups are multiplied or the composite of two elements of the set of the group, then the new element formed should be of the same set.
II) The law of associativity is satisfied under this composition.
III) Given a common composition of two elements and a second composition where there are only three elements, and no twoidentical elements are being composed; then the two elements that appearonce must be equivalent.
IV) A group is said to be Abelian if the composite of A*B=B*A.
Weber also proves several logical relationships between composites of the elements of the group.
To show off his approach to abstract algebra, Weber said: "One can combine all groups isomorphicto one another into a class of groups, which itself is again a group, whose elements are the generic characters which one obtains if one concept, and it is irrelevant which representative on uses to study the properties of the group" (Katz 609). He then goes on to show how vector spaces under vector addition are a group, multiplication group of residues classes under mod m relatively prime to m are groups, etc.
Sophus Lie is the last part of the story on groups. Sophus Lie determined the necessary criterion for sets with binary operations or permutations to satisfy.

The criterion for a set (M) with a binary operation (+) to be a group must satisfy four properties. Given a, b and c to be elements of M then
1) a+b=d where d must be in the group {this is known as the closure property}
2) a+(b+c)=(a+b)+c {Commutativity}
3) there must exist and element e such that a+e=e+a=a{identity}
4) for a in M then there must exist a wherea+a=a+a=e {inverse}A fifth property for certain groups say that a+b=b+a,this group is called an abelian group. As was stated above, given any set of permutations then it must be a group under function composition.
As mentioned earlier, group theory has many practical applications in physics, real analysis, and other areas of mathematics. In physics, group theory is employed in quantum mechanics and classical mechanics. In quantum mechanics, the development of the space on which all of the operators are based and used. The first chapter of just about every quantum mechanics book,Melissinos, Dirac, etc(see bibliography); is the development of the vector space and applications on state vectors, which can be used to represent an electron configuration in an atom or a free particle, etc. As Weber had shown these form a group under vector addition, that these vectors when added add component wise. As shown in Wigners group theory, the Slater determinants, determinants of state functions ?(I), used in the calculations of energies and the Hartree-Fock equations for electrons in an atom. The matrix formed from these matrix elements is apart of a group of matrices. Any of the symmetries used in physics, e.g.-spherical harmonics for a filled electron shell, can be represented asa multiplication table of transformations or a Cayley table described above. A Minkowski space has been shown to be a group, a Minkowski space is used pre- eminently in special relativity and is the coupling of time to the spatial vectors which leaves time transforming not independently of spatial components but identically. In classical mechanics, time was considered as most lay people consider time, i.e. constant through out the cosmos; but in fact time has been shown to transform from one coordinate frame to another in a similar way to the spatial components.
In mathematics, group theory can be altered into a generalized field theory or Galois Theory. All that is required for a field is an abelian group with a second binary operation that contains a unity. Essentially the set of the abelian group minus the identity e must be an abelian group with respect to the second binary operation.
The story of group theory is far deeper than is apparent in most modern algebra texts. Modern Algebra is really a unity of three "sub- fields" of mathematics, namely group theory, field thoery, and geometry. It turns out that calculus itself is based on group theory. A well-defined field is necessary for the beginnings of naïve set theory to build up the necessary theorems for calculus. This work was done in the 19th century by the likesof Riemann, Cauchy, Lebesgue, etc. It is this fundamental study of the entire base of mathematics.