Theory of Finite Groups by William Burnside is among the most modern books I helped digitize at Distributed Proofreaders (DP) for Project Gutenberg. Brenda Lewis, for whom I served as Post-Processor on a number of projects, was Content Provider and Project Manager. As with almost all books produced at DP, unnamed volunteers did the skilled, painstaking work of correcting OCR output one page at a time.
In mathematics, a group is a type of algebraic structure that axiomizes the concept of symmetry. From this perspective, elements of a group are transformations; think “shift to the left by 5mm” or “rotate counterclockwise 2 degrees about the center” as operations on a photocopier original. These operations are elements of the group of proper plane motions, transformations preserving distance and signed angle, for which clockwise and counterclockwise differ.
If in addition we include transformations such as “reflect left to right about the center line” or “enlarge by 200% relative to the upper left corner,” we obtain other, larger groups of plane transformations. Not all groups consist of plane transformations. Just for example, a group may act as “symmetries” on another group.
More formally, a group consists of a non-empty set \(G\) and a “group operation” on \(G\), a way of “multiplying” two elements \(g\) and \(g’\) in \(G\) to get an element \(gg’\) in \(G\). The group operation satisfies three axioms:
First, the group operation is associative: For all \(g\), \(g’\), and \(”\) in \(G\), we have \((gg’)g” = g(g’g”)\). Plane transformations satisfy this condition automatically.
Second, the group operation has an identity element \(e\) in \(G\), namely, satisfying \(eg = g\) and \(ge = g\) for all \(g\) in \(G\). The plane transformation “stay put” acts as identity element; indeed, mathematicians call this the identity mapping.
Third, every group element has an inverse: For every \(g\) in \(G\), there exists a group element \(g’\) such that \(gg’ = e\) and \(g’g = e\). Operationally, every “symmetry” can be undone, and undoing is itself a symmetry.
Burnside’s treatise concerns groups having finitely many elements. For example, a square in the plane has a finite symmetry group: There are precisely eight plane motions that map the square to itself. So much can be said about finite groups, however, that students do not normally see much of the material in Burnside until late undergraduate or early graduate study, and then only if the material impinges on individual specialization.
These brief remarks about groups illustrate deep, important facts about the structure of mathematics and our own human limitations: The definition of a group can be given easily, little more than what is stated above. The logical consequences of this definition suffice to fill an introductory book of several hundred pages, and far more, a web of concepts and theorems extending beyond human knowledge and understanding.