Simple groups and a Diophantine equation

Leo James Alex

Vol. 104 (1983), No. 2, 257–262

DOI: 10.2140/pjm.1983.104.257

Abstract

Let G be a finite simple group whose order is of the form pm where p is a prime, (p,m) = 1, and the index of a Sylow p-subgroup in its normalizer is three in G. Suppose the degree equation for the principal p-block, B₀(p), has the form 1 + 2^a = 3^b5^c + 2^d3^e5^f where a, b, c, d, e and f are non-negative integers. In this paper it is shown that under these conditions G must be isomorphic to one of the groups L(2,7), U(3,3), L(3,4) and A₈. This is accomplished by solving the exponential Diophantine degree equation for B₀(p).

Mathematical Subject Classification 2000

Primary: 20D05

Secondary: 10B25

Milestones

Received: 9 September 1980

Published: 1 February 1983