Ferdinand Georg Frobenius - Contributions To Group Theory

Contributions To Group Theory

Group theory was one of Frobenius' principal interests in the second half of his career. One of his first contributions was the proof of the Sylow theorems for abstract groups. Earlier proofs had been for permutation groups. His proof of the first Sylow theorem (on the existence of Sylow groups) is one of those frequently used today.

• Frobenius also has proved the following fundamental theorem: If a positive integer n divides the order |G| of a finite group G, then the number of solutions of the equation xn = 1 in G is equal to kn for some positive integer k. He also posed the following problem: If, in the above theorem, k = 1, then the solutions of the equation xn = 1 in G form a subgroup. Many years ago this problem was solved for solvable groups. Only in 1991, after the classification of finite simple groups, was this problem solved in general.

More important was his creation of the theory of group characters and group representations, which are fundamental tools for studying the structure of groups. This work led to the notion of Frobenius reciprocity and the definition of what are now called Frobenius groups. A group G is said to be a Frobenius group if there is a subgroup H < G such that

for all .

In that case, the set

together with the identity element of G forms a subgroup which is nilpotent as Thompson showed in his PhD thesis. All known proofs of that theorem make use of characters. In his first paper about characters (1896), Frobenius constructed the character table of the group of order (1/2)(p3 − p) for all odd primes p (this group is simple provided p > 3). He also made fundamental contributions to the representation theory of the symmetric and alternating groups.