HOW GROUP THEORY CAME ABOUT
The multiplicity of various
kinds of algebraic notions invented in the 19th century might have
given mathematics a centrifugal tendency had it not been for the development of
structural concepts. One of the most important of these was the notion of a
group. In algebra it was undoubtedly the most important force making for
cohesiveness. The group-property is a common bond almost coextensive with
modern algebra as a whole. Now the entire algebra is rich with group theoretic
techniques.
The history of group theory is old
and rich. Although the concept of group and its name is a creation of the 19th
century, its roots can be traced back to the Babylonian civilization in the 5th
to 4th century BC when mathematicians strived to find a general
method of solution for finding roots of a linear equation. Although they did
not have the notion of an equation, they did develop an algorithmic approach to
solving problems, which, in our terminology, would give rise to a linear
equation. In E.Galois' words, group theory is a "metaphysics of theory of
equations".1
With
the passage of time the struggle to find a general method for solutions of
algebraic equations of degrees higher than one continued. Almost all the great
mathematicians had ventured to devise a general method for finding solutions of
equations of all degrees until the 19th century when the struggle
took a sharp turn with the advent of the famous result, the insolubility of
equations of degree higher than four, by J.L.Lagrange (1735-1813) and N.H.Abel
(1802-1829).
The
first ever general method of finding solutions of the quadratic equation did
not appear until the time of Mohammad ibn-i-Musa al-Khowarzmi (c.790-850).
Historical records show the evidence of the existence of a method of finding
one solution of the quadratic equation by Brahamagupta (598-665). Al-Khowarzmi
is known to be the first mathematician to have used non-geometrical means to
construct quadratic equations and give a general method of solving them for
positive roots. The terms-wise classification of all linear and quadratic
equations having positive roots and the general solutions for each type are
available in chapters one to six of his well-known book, al-jabr wal muqabalah.3 His work had a tremendous impact
on later mathematicians. One of those was Abraham bar Hiyya Ha-Nasi
(c.1065-c.1136), commonly known as Savasorda, who is well known for his book Liber Embadorum published in 1145. His
was the first book published in Europe to give the complete solution of the
quadratic equation.
Mathematicians in this golden age of
scholarship were all concerned mainly with the algebra of al-Khowarzmi,
studying solutions of algebraic equations. Al-Mahani (c.860) was the first to
have considered cubic equation of a special type. One of his contemporaries,
Thabit ibn-Qurra (826-901), extended his work to some more special cases of
cubic equations. Although they did not
contribute significantly to the general algebraic theory in the solution of the
cubic, they were perhaps the first ones to solve the cubic by geometric
methods. It was later that Omar Khayyam (1048-1122) was the first to classify
the cubic equations systematically and to obtain a root as the abscissa of a
point of intersection of a circle and a rectangular hyperbola, or of two
rectangular hyperbolas. He was also the one to have handled various types of
cubic equations that possess positive roots.
Besides
the cubic equations, mathematicians also took up the quartic equations,
although like the cubic equations they seemed to have managed only with
geometric methods and with positive numbers as their roots. The quartic
equations were not as general as we see in later years. Abu-Wafa (940-998), who
had written a commentary on al-jabr wal
muqabalah, gave geometric solutions to some special quartic equations.
However, his work on such problems is lost but reference to this is found in
Abu-Faradsh Mohammad ibn-Ishaq’s book entitled Kitab al-Fihrist (c.987). Thus, by the 12th century,
general methods for finding solutions of linear and quadratic equations had
existed and also there is evidence of the existence of general geometric
methods for finding solutions of special types of cubic as well as quartic
equations.
The
struggle to find general methods for finding solutions of cubic and quartic
equations continued. Medieval European mathematicians made sporadic attempts
also. L.Fibonacci (1170-1250) was one of those. It was not until the
Renaissance period (1450 to 1700) that general algebraic methods for finding
solutions of cubic and quartic were found. Italian mathematicians, namely
Scipione dal Ferro (c.1465-1526), N.Tartaglia (c.1500-1557) and G.Cardano
(1501-1576) were at last successful in finding a general method§ to
find solutions of the general cubic equation. Later, L.Ferrari (1522-1565) devised
an algebraic method for finding solutions of the most general quartic equation.
The
struggle to devise general algebraic method to find solutions of equations
beyond quartic equations continued but in vain. Later in the 18th
century, L.Euler (1707-1783) first worked on the idea that the problem of
quintic equations could be reduced to that of solving quartic equations.
J.L.Lagrange made the same attempt. In 1767 appeared his memoir Sur la Resolution des Equations Numeriques
in which he described methods of separating the real roots of an algebraic
equation and of approximating them by means of continued fractions. This was
followed in 1770 by the Reflexions sur la
Resolution Algebrique des Equations that dealt with the fundamental
question of why the methods useful to solve linear, quadratic, cubic and
quartic are not successful for solving quintic equations. This led Lagrange to
see the importance of rational functions of the roots and their behaviour under
the permutations of the roots. This inspired the works of P.Ruffini (1765-1822)
and N.H.Abel on quintic equations, and at last in a pamphlet published in 1824,
N.H.Abel proved the impossibility of solving the general quintic equation by
means of radicals.
N.H.Abel and P.Ruffini first attempted to prove that the
quintic equation could not be solved by radicals. A.I.Cauchy (1789-1857)
studied the group of permutations of roots of equations of higher orders for
its own interest but the complete description of the relationship between
groups and algebraic equations was first given by E.Galois (1811-1832) 4.
N.H.Abel’s work inspired young E.Galois. He expressed the fundamental
properties of the transformation group belonging to the roots of an algebraic
equation and showed that the field of rationality of these roots was determined
by the group. He thus discovered that an irreducible algebraic equation is
soluble by radicals if and only if a certain group of permutations of its roots
is soluble. In the words of D.J.Struik: Galois' unifying principle is now
considered as one of the outstanding achievements of 19th century
mathematics.3 Philip Hall
compared the historical role of the quintic equation in algebra with the axiom
of parallels in geometry.2
The
importance of group theory is not limited to this special problem of finding
solution of a quintic equation by the same methods which were successful for
cubic and quartic equations. Later, C.Jordan developed a detailed exposition of
the theory due to N.H.Abel and E.Galois. He published the first lasting
monument of these ideas, namely Traite
des Substitutions et des Equations in 1870. E.Galois gave the name group
but A.Cayley (1821-1895) and L.Kronecker (1823-1891) later defined it
axiomatically. From this time, group theory took an explicit form and has since
played a fundamental role in all fields of mathematics.
The theory of groups by this time had developed to a
great extent into a systematic theory. Its scope started widening. In P.Hall's
words: 'It is not stretching too far to say that any type of algebra of any
degree of importance is in some sense or the other related to group theory'.2
Its linkage with other branches of mathematics started appearing. P.De Fermat
(1601-1665) and C.F.Gauss (1777-1855) established its link with Number theory.
The significance of this linkage can be seen from the fact that the number
systems, such as those of integers, real numbers, complex numbers, etc., are
groups.
F.Klein
(1849-1925) emphasized the significance of group theory in geometry. In his
inaugural address at the eve of his professorial appointment at Erlangen, he
explained the importance of the group conception for the classification of the
different fields of mathematics. The address, which became known as the Erlangen Program, declared every geometry
to be the theory of invariants of a particular transformation group. He pointed
out that by enlarging or contracting the group one could pass from one type of
geometry to another. That is, classification of groups of transformations
yields a classification of geometry.4 Every symmetrical object of
whatever kind can be associated with a group. It is on this association that
the application of groups in geometry and physics largely depends.
This
linkage was later given another dimension by M.S.Lie (1842-1899). He developed
the theory of Lie groups in the 1880s by imposing a topological structure on
group. He had discovered the contact transformation and with this the key to
the entire Hamiltonion dynamics as a part of group theory. This led him to
study continuous transformation groups and their invariants, demonstrating
their central importance as a classifying principle in geometry, mechanics,
ordinary and partial differential equations.
The
relationship between group theory and physics became significantly strong and
concrete when H.Weyl (1885-1955) published his famous book Gruppentheorie und Quantenmechanik in 1928. He evolved in it, a general theory of
continuous groups using matrix representation and found that most of the
regularities of quantum phenomena on the atomic level can be most simply
understood using group theory. The theory of groups made possible a synthesis
of the geometrical and algebraic work of G.Monge (1746-1818), V.Poncelet
(1788-1867), C.F.Gauss, A.Cayley, A.Clebsch (1833-1872), H.Grassmann
(1809-1877), and B.Reimann (1826-1866). The invention of group theory therefore
occupies a fundamentally central position in mathematics.
References
1.
L.E.Dickson, Linear Groups: With an Exposition of the Galois Field Theory, 1958,
Dover Publications, N.Y.
2.
P.Hall, What
Modern Algebra is About, Eureka, 3(1940),
12-14.
3.
Q.Mushtaq and A.L.Tan, Mathematics: The Islamic Legacy, Centre of Central Asian Studies
(UNESCO), Islamabad, 1993.
4. D.J.Struik,
A Concise History of Mathematics,
1962, Dover Publications, N.Y.
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