Simple Groups of Order 2ᵃqᵇr²

Citation

Leon, Jeffrey Samuel (1971) Simple Groups of Order 2ᵃqᵇr². Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/d9kz-zp57. https://resolver.caltech.edu/CaltechTHESIS:02242025-160744993

Abstract

In this thesis we study simple groups of order p^aq^br² (p, q, and r prime numbers). John Thompson has shown that, in any simple {p,q,r}-group, the primes dividing the group order are 2, 3, and an element of {5, 7, 13, 17}. Richard Brauer and David Wales have classified simple groups of order p^aq^br. In the case of interest here, known results permit us to write the group order as 2^aq^br² unless the group is isomorphic to A₅. We shall deal primarily with the case r = 3.

Let G be a simple group with |G| = 2^aq^b3². Recent work of W. J. Wong on the relation between blocks and exceptional characters provides the key to obtaining information about the principal 3-block of G. Using the block-section orthogonality relations, we obtain diophantine equations for the degrees of the irreducible characters in this block. Methods are developed for solving the type of equation which arises. Finally, we perform a detailed analysis of the solutions; the most important technique is restriction of characters to 3-local subgroups.

Let k₃(G) denote the number of conjugacy classes of 3-elements in G. It is proven that, if kk₃(G) ≥ 3, then G must be isomorphic to one of the linear groups PSL₂(8) or PSL₂(17). If k₃(G) = 2, either G is isomorphic to the alternating group A₆ or else the principal 3-block of G has one of three explicit forms. Results for the case k₃(G) = 1 are weaker; however, they permit us to show that |G| ≠ 2^aq²3² and that, with known exceptions, |G| > 10⁶. The latter result serves to prove the nonexistence of a simple group for a number of previou3ly unresolved orders.

In deriving the above theorems, we establish a number of preliminary results valid in any simple group of order 2^aq^br². We also prove a number of theorems on 3-blocks of defect 2 and groups of order 3²t, (3,t) = 1.

Thesis Files