Ferdinand Georg Frobenius was born to a Protestant minister in Berlin. Having graduated from school at 18, he had a short spell at the University of Göttingen, but soon returned to Berlin. He enrolled at the University there, and began his studies under the eminent mathematicians Kronecker, Kummer and Weierstrass. In 1870 he received his doctorate which was awarded with distinction!
Frobenius then began to teach secondary students at his old school Joachimsthal Gymnasium (Gymnasium is the name for a Grammar School), before being offered an appointment as extraordinary professor in mathematics at his old University.
Frobenius did eventually move away from Germany to take up the job of professor at the Polytechnic of Zurich (Switzerland) - another German speaking area.
He stayed there for over 15 years, where he married and raised a family. He returned to Berlin after the death of his old professor Kronecker and with some help from Weierstrass he took over the vacant chair.
This was to be the start of an interesting period for Frobenius, the university, and also the development and direction of modern-day mathematics.
Frobenius was of the old school of thought (a traditionalist) in terms of the study of “pure” mathematics. He felt that ‘applied’ mathematics was a lesser discipline not suited to a world-class university. The new mathematics being developed at the time by the likes of Fuchs - applied mathematics - was in his opinion “something that should be taught at the (inferior) technical colleges” like the University of Göttingen.
The irony is that Göttingen flourished as a world centre of excellence for applied mathematics, which really irritated Frobenius. Despite his elitist belief, there was a slow decline in academic achievements at the University of Berlin.
Not everything was a negative for Frobenius and Berlin. Frobenius’ most remembered contribution to mathematics, in group theory, was to have another twist for modern-day mathematics.
His fundamental work in the theory of finite groups was later to have important implications in quantum mechanics and theoretical physics - the applied mathematical areas that in his time he had complete and utter disdain for.
Groups are a subject that emerged during the nineteenth century as a fundamental part of mathematics. They explain and unify many areas that had traditionally caused problems. Under the fundamental contributions of Abel, Lagrange and Galois they provided an explanation about the solvability of equations, and in particular explained why quintic and higher degree equations could not be solved by radicals.
But what is a group? Consider an equilateral triangle lying in a plane, and the number of ways that it may be picked up and replaced so that there is no discernible difference between the ‘before’ and the ‘after’. These are the symmetries of the triangle. There are just six of them - three reflections about axes (like the one shown below, r), and three rotations (like the one shown below, s). Together, they make up a group. We will consider one of each type:
Under the reflection r, the vertex A remains fixed, but B and C and C and B are interchanged. We may write,
Under the rotation s, we have,
But we may combine the symmetries, so that when we perform s and then r (in that order) we have:
notice that this is another reflection, but through the different axis shown. Similarly we have:
this is a different reflection, this time about a third such axis:
Any group has four basic properties:
Notice that in the group of symmetries of the equilateral triangle,
We say that it is non-abelian. Groups which for all elements r and s,
are called abelian.
The symmetries correspond to permutations of the vertices, and they have their own ‘artithmetic’. Equations have similar symmetries and corresponding permutations concerning their roots. The arithmetic associated with them (in either case) is called group theory, and the structure of such groups determines the behaviour of the objects that create them.