On Burnside's method
This paper considers the following theorem of William Burnside: If a primitive permutation group \(G\) contains a transitive cyclic subgroup of prime power order \(p^m\) with \(m>1\), then \(G\) is doubly transitive. This theorem and related results were given by Burnside in the 1911 edition of his well-known group theory book (JFM 42.0151.02), using the character theory (over the complex numbers) of the permutation module of the group \(G\). However, Burnside's proof of this theorem was incorrect, as pointed out by \textit{P. M. Neumann} [in ``Helmut Wielandt, Mathematische Werke'' (de Gruyter, Berlin, 1994; Zbl 0802.20001), Vol. I, 3-20]. Burnside's proof was in some sense superseded by a 1933 proof given by Schur and the subsequent development of Schur-ring theory. But the present paper gives a corrected version of Burnside's proof based on Burnside's ideas, and also points out the close connection between Burnside's method and Schur's method of dealing with this kind of question. Several propositions are included about the more general situation of a permutation group \(G\) with a transitive (regular) abelian subgroup. The method is based on considering the ``Burnside basis'' (as well as the ``Schur basis'') for the transitivity submodule (fixed by the point-stabilizer subgroup) in the permutation module of \(G\) (over a sufficiently large field of characteristic 0). When \(G\) is primitive and the regular subgroup is cyclic of order \(p^m\), this leads to a relation among certain roots of unity. This relation does not imply precisely what Burnside claimed it did, but an elementary number-theoretic fact about roots of unity is used to complete the corrected proof of Burnside's theorem.
scientific article; zbMATH DE number 783941