Group theory: v v v brief historical origins

Reference: Abstract Algebra, Second Edition, Dummit and Foote. John Wiley and Sons Inc.

The modern treatment of abstract algebra begins with the disarmingly simple abstract definition of a group. This simple definition quickly leads to difficult questions involving the structure of such objects. There are many specific examples of groups and the power of the abstract point of view becomes apparent when results for all of these examples are obtained by proving a single result for the abstract group.

The notion of a group did not simply spring into existence, however, but is rather the culmination of a long period of mathematical investigation, the first formal definition of an abstract group in the form in which we use it appearing in 1882. The definition of an abstract group has its origins in extremely old problems in algebraic equations, number theory, and geometry, and arose because very similar techniques were found to be applicable in a variety of situations. As Otto Holder (1859-1937) observed, one of the essential characteristics of mathematics is that after applying a certain algorithm or method of proof one then considers the scope and limits of the method. As a result, properties possessed by a number of interesting objects are frequently abstracted and the question raised: can one determine all the objects possessing these properties ? Attempting to answer such a question also frequently adds considerable understanding of the original objects under consideration. It is in this fashion that the definition of an abstract group evolved into what is, for us, the starting point of abstract algebra.

We illustrate with a few of the disparate situations in which the ideas later formalized into the notion of an abstract group were used.

  1. In number theory the very object of study, the set of integers, is an example of a group. Consider, for example, what we refer to as “Euler’s Theorem”, one extremely simple example of which is that a^{40} has last two digits 01 if a is any integer not divisible by 2 nor by 5. This was proved in 1761 by Leonhard Euler (1707-1783) using “group-theoretic” ideas of Joseph Louis Lagrange (1736-1813), long before the first formal definition of a group. From our perspective, one now proves “Lagrange’s Theorem”, applying these techniques abstracted to an arbitrary group, and then recovers Euler’s theorem (and many others) as a special case.
  2. Investigations into the question of rational solutions to algebraic equatios of the form y^{2}=x^{3}-2x (there are infintely many, for example (0,0), (-1,1), (2,2), (\frac{9}{4}, \frac{-21}{8}), (\frac{-1}{169}, \frac{239}{2197})) showed that connecting any two solutions by a straight line and computing the intersection of this line with the curve y^{2}=x^{3}-2x produces another solution. Such “Diophantine equations,” among others, were considered by Pierre de Fermat (1601-1655) (this one was solved by him in 1644),by Euler, by Lagrange around 1777, and others. In 1730 Euler raised the question of determining the indefinite integral \int (\frac{dx}{\sqrt{1-x^{2}}}) of the “lemniscatic differential” \frac{dx}{\sqrt{1-x^{2}}}, used in determining the arc length along an ellipse (the question had also been considered by Wilhelm Gottfried Leibniz (1646-1716) and Johannes Bernoulli (1667-1748)). In 1752 Euler proved a “multiplication formula” for such elliptic integrals (using ideas of G. C. di Fagnano (1682-1766), received by Euler in 1751), which shows how two elliptic integrals give rise to a third, bringing into existence the theory of elliptic functions in analysis. In 1834 Carl Gustav Jacob Jacobi (1804-1851) observed that the work of Euler on solving certain Diophantine equations amounted to writing the multiplication formula for certain elliptic integrals. Today the curve above is referred to as an “elliptic curve” and these questions are viewed as two different aspects of the same thing — the fact that this geometric operation on points can be used to give the set of points on an elliptic curve the structure of a group. The study of the “arithmetic” of these groups is an active area of current research.
  3. By 1824 it was known that there are formulas giving the roots of quadratic, cubic and quartic equations (extending the familiar quadratic formula for the roots of ax^{2}+bx+c=0). In 1824, however, Niels Henrik Abel (1802-1829) proved that such a formula for the roots of a quintic is impossible. The proof is based on the idea of examining what happens when the roots are permuted amongst themselves (for example, interchanging two of the roots). The collection of such permutations has the structure of a group (called, naturally enough, a “permutation group”). This idea culminated in the beautiful work of Evariste Galois (1811-1832) in 1830-32, working with explicit groups of “substitutions.” Today this work is referred to as Galois Theory. Similar explicit groups were being used in geometry as collections of geometric transformations (translations, reflections, etc) by Arthur Cayley (1821-1895) around 1851. Camille Jordan (1838-1922) around 1867, Felix Klein (1849-1925) around 1870 etc. and the application of groups to geometry is still extremely active in current research into the structure of 3-space, 4-space, etc. The same group arising in the study of the solvability of the quintic arises in the study of the rigid motion of an icosahedron in geometry and in the study of elliptic functions in analysis.
  4. The precursors of todays group can be traced back many years, even before the groups of “substitutions” of Galois. the formal definitionof an abstract group which is our starting point appeared in 1882 in the work of Walter Dyck (1856-1934), an assistant to Felix Klein, and also in the work of Heinrich Weber (1842-1913) in the same year. It is freuently the case in mathematics research to find specific application of an idea before having that idea extracted and presented as an item of interest in its own right (for example, in 1830 Galois used the notion of quotient group implicitly in his investigations in 1830 and the definition of an abstract quotient group is due to Holder in 1889). It is important to realize, with or without the historical context, that the reason the abstract definitions are made is because it is useful to isolate specific characteristics and consider what structure is imposed on an object having these characteristics. The notion of the structure of an algebraic object (which is made more precise by the concept of an isomorphism —- which considers when two apparently different objects are in some sense the same) is a major theme that recurs in algebra.

Hope you enjoyed !

Cheers,

Nalin Pithwa

Generalized associative law, gen comm law etc.

Reference : Algebra by Hungerford, Springer Verlag, GTM.

Let G be a semigroup. Given a_{1}, a_{2}, a_{3}, \ldots, a_{n} \in G, with n \geq 3, it is intuitively plausible that there are many ways of inserting parentheses in the expression latex a_{1}a_{2}\ldots a_{n}$ so as to yield a “meaningful” product in G of these n elements in this order. Furthermore, it is plausible that any two such products can be proved equal by repeated use of the associative law. A necessary prerequisite for further study of groups and rings is a precise statement and proof of these conjectures and related ones.

Given any sequence of elements of a semigroup G, (a_{1}a_{2}\ldots a_{n}) define inductively a meaningful product of a_{1}, a_{2}, \ldots, a_{n} (in this order) as follows: If n=1, the only meaningful product is a_{1}. If n>1, then a meaningful product is defined to be any product of the form (a_{1}\ldots a_{m})(a_{m+1}\ldots a_{n}) where m<n and (a_{1}\ldots a_{m}) and (a_{m+1} \ldots a_{n}) are meaningful products of m and n-m elements respectively. (to show that this statement is well-defined requires a version of the Recursion Theorem). Note that for each n \geq 3 there may be meaningful products of a_{1}, a_{2}, \ldots, a_{n}. For each n \in \mathcal{N}^{*} we single out a particular meaningful product by defining inductively the standard n product \prod_{i=1}^{n}a_{i} of a_{1}, \ldots, a_{n} as follows:

\prod_{i=1}^{1}a_{i}=a_{1}, and for n>1, \prod_{i=1}^{n}a_{i}=(\prod_{i=1}^{n-1})a_{n}

The fact that this definition defines for each n \in \mathcal{N}^{*} a unique element of G (which is clearly a meaningful product) is a consequence of the Recursion Theorem.

Theorem: Generalized Associative Law:

If G is a semigroup and a_{1}, a_{2}, \ldots, a_{n} \in G, then any two meaningful products in a_{1}a_{2}\ldots a_{n} in this order are equal.

Proof:

We use induction to show that for every n any meaningful product a_{1}a_{2} \ldots a_{n} is equal to the standard n product \prod_{i=1}^{n}a_{i}. This is certainly true for n=1, 2. For n>2, by definition, a_{1}a_{2} \ldots a_{n} = (a_{1}a_{2}\ldots a_{m})(a_{m+1} \ldots a_{n}) for some m<n. Therefore by induction and associativity:

(a_{1}a_{2} \ldots a_{n}) = (a_{1}a_{2} \ldots a_{m})(a_{m} \ldots a_{n}) = (\prod_{i=1}^{m}a_{i})(\prod_{i=1}^{n-m})a_{m+i}

= (\prod_{i=1}^{m}a_{i}) ((\prod_{i=1}^{n-m-1}a_{m+i})a_{n} ) = ((\prod_{i=1}^{m})(\prod_{i=1}^{n-m-1}a_{m+i}))a_{n} = (\prod_{i=1}^{n-1}a_{i})a_{n} = \prod_{i=1}^{n}a_{i}

QED.

Corollary: Generalized Commutative Law:

If G is a commutative semigroup and a_{1}, \ldots, a_{n}, then for any permutation i_{1}, \ldots, i_{n} of 1, 2, …,n a_{1}a_{2}\ldots a_{n} = a_{i_{1}}a_{i_{2}}\ldots a_{i_{n}}

Proof: Homework.

Definition:

Let G be a semigroup with a \in G and n \in \mathcal{N}^{*}. The element a^{n} \in G is defined to be the standard n product \prod_{i=1}^{n}a_{i} with a_{i}=a for 1 \leq i \leq n. If G is a monoid, a^{0} is defined to be the identity element e. If G is a group, then for each n \in \mathcal{N}^{*}, a^{-n} is defined to be (a^{-1})^{n} \in G.

It can be shown that this exponentiation is well-defined. By definition, then a^{1}=a, a^{2}=aa, a^{3}=(aa)a=aaa, \ldots, a^{n}=a^{n-1}a…and so on. Note that it is possible that even if n \neq m, we may have a^{n} = a^{m}.

Regards.

Nalin Pithwa

A non trivial example of a monoid

Reference : Algebra 3rd Edition, Serge Lang. AWL International Student Edition.

We assume that the reader is familiar with the terminology of elementary topology. Let M be the set of homeomorphism classes of compact (connected) surfaces. We shall define an addition in M. Let S, S^{'} be compact surfaces. Let D be a small disc in S, and D^{'} in S^{'}. Let C, C^{'} be the circles which form the boundaries of D and D^{'} respectively. Let D_{0}, D_{0}^{'} be the interiors of D and D^{'} respectively, and glue S-D_{0} to S^{'}-D_{0}^{'} by identifying C with C^{'}. It can be shown that the resulting surface is “independent” up to homeomorphism, of the various choices made in preceding construction. If \sigma, \sigma_{'} denote the homeomorphism classes of S and S^{'} respectively, we define \sigma + \sigma_{'} to be the class of the surface obtained by the preceding gluing process. It can be shown that this addition defines a monoid structure on M, whose unit element is the class of the ordinary 2-sphere. Furthermore, if \tau denotes the class of torus, and \pi denotes the class of the projective plane, then every element \sigma of M has a unique expression of the form

\sigma = n \tau + m\pi

where n is an integer greater than or equal to 0 and m is zero, one or two. We have 3\pi=\tau+n.

This shows that there are interesting examples of monoids and that monoids exist in nature.

Hope you enjoyed !

Regards,

Nalin Pithwa

Algebra is symbolic manipulation though painstaking or conscientious :-)

Of course, I have oversimplified the meaning of algebra. 🙂

Here is an example. Let me know what you think. (Reference: Algebra 3rd Edition by Serge Lang).

Let G be a commutative monoid, and x_{1}, x_{2}, \ldots, x_{n} be elements of G. Let \psi be a bijection of the set of integers (1,2, \ldots, n) onto itself. Then,

\prod_{v=1}^{n}x_{\psi(v)} = \prod_{v=1}^{n}x_{v}

Proof by mathematical induction:

PS: if one gets scared by the above notation, one can expand it and see its meaning. Try that.

It is clearly true for n=1. We assume it for n=1. Let k be an integer such that \psi(k)=n. Then,

\prod_{i}^{n}x_{\psi(v)} = \prod_{1}^{k-1}x_{\psi(v)}.x_{\psi(k)}.\prod_{1}^{n-k}x_{\psi(k+v)}

= \prod_{1}^{k-1}x_{\psi(v)}. \psi_{1}^{n-k}x_{\psi(k+v)}.x_{\psi(k)}

Define a map \phi of (1,2, \ldots, n-1) into itself by the rule:

\phi(v)=\psi(v) if v<k

\phi(v) = \psi(v+1) if v \geq k

Then,

\prod_{1}^{n} x_{\psi(v)} = \prod_{1}^{k-1}x_{\phi(v)}. \prod_{1}^{n-k}x_{\phi(k-1+v)} = \prod_{1}^{n-1}x_{\phi(v)}.x_{n}

which by induction is equal to x_{1}\ldots x_{n} as desired.

Some remarks: As a student, I used to think many a times that this proof is obvious. But it would be difficult to write it. I think this cute little proof is a good illustration of “how to prove obvious things in algebra.” 🙂

Regards,

Nalin Pithwa

Wisdom of Hermann Weyl w.r.t. Algebra

Important though the general concepts and propositions may be with which the modern and industrious passion for axiomatizating and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity consitute the stock and core of mathematics, and that to master their difficulties requires on the whole hard labour.

—- Prof. Hermann Weyl.