Several recent problems in the representation theory of finite groups require determining whether certain characters of almost simple groups belong to the principal block. Since the values of these characters are not yet known, we employ alternative group-theoretical techniques to address the "going down" case. This approach enables us to reduce the block version of well-known results by the third and fourth authors to a question about almost simple groups. Moreover, this suggests a Galois analogue of the height-zero-equal-degree conjecture of Malle and Navarro, which we formulate. However, the "going up" case of irreducible extensions of principal block characters remains unresolved.

This paper highlights the use of transfer to analyze an unbalancing configuration in the GLS classification project for finite simple groups. The configuration occurs when e(G) = 3, m2,p(G) = mp(G) = 3 for some odd prime p, and a p-component of p-rank 1 is responsible for the unbalancing.

General introduction to the special odd case General lemmas Theorem $C^*_2$: Stage 1 Theorem $C^*_2$: Stage 2 Theorem $C_2$: Stage 3 Theorem $C_2$: Stage 4 Theorem $C_2$: Stage 5 Theorem $C_3$: Stage 1 Theorem $C_3$: Stages 2 and 3 IV$_K$: Preliminary properties of $K$-groups Background references Expository references Glossary Index.

If G is a finite group, p is a prime, and x is an element of G, it is an interesting problem to place x in a convenient small (normal) subgroup of G, assuming some knowledge of the order of the products xy, for certain p-elements y of G.

In a burst of activity between the late 1950’s and the early 1980’s, one of the biggest mathematical stories of the twentieth century was told—that of the classification of the finite simple groups. The peerless leader of the analysis of arbitrary finite simple groups was John Griggs Thompson, from the moment he came on the scene. His vision lit the entire enterprise, while his inventions supplied and inspired brand-new mathematical tools. The Thompson chapter surveys his contributions to finite group theory as well as Galois theory and other areas of algebra.

Among the simplest invariants of the sporadic finite simple groups are their outer automorphism groups. For 12 of the 26 possible isomorphism types of a sporadic simple group G, the outer automorphism group Out(G) has order 2, and in the remaining 14 cases, Out(G) is trivial. Historically the suspicion of the existence of a sporadic group was followed in fairly short order by the calculation of a good upper bound on the size of its outer automorphism group. In a few cases establishing the existence of certain outer automorphisms, like the existence of the groups themselves, presented difficulties overcome only with the use of machine computation. In any case these calculations can be difficult to track down in the literature. This note, which contains nothing new, is only intended to bring together these calculations. It also answers a question of Bob Oliver about the automorphism groups of some of these groups, how they might be calculated, and specifically whether the Sylow 2-subgroups of a sporadic simple group are self-centralizing in the automorphism group of the simple group. The answer is that they are.