For a finite group G let (G) be the (simple) graph defined on the elements of G with an edge between two (distinct) vertices if and only if they generate G. The chromatic number of (G) is considered for various non-solvable groups G.

the theorem 12 in [a note on $p$-nilpotence and solvability of finite groups, j. algebra 321(2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indice. in this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index aresolvable.

In a recent paper [1], Adem, Maginnis and Milgram calculated the mod 2 cohomology of the Mathieu simple group M12. An interesting feature of the answer they obtain is that it is Cohen–Macaulay. There is a subalgebra isomorphic to the rank three Dickson invariants, namely a polynomial ring in generators of degrees 4, 6 and 7, and over this subalgebra the whole cohomology ring is a finitely gener...

This paper represents the first step in the classification of finite primitive distance transitive graphs. In it we reduce the problem to the case where the automorphism group is either almost simple or affine. Let ^ be a simple, connected, undirected graph with vertex set Q. If oc, /? e Q, then d(a, j8) denotes the distance between a and /3 in §. Let G be some group of automorphisms of §. Then...

[UMSt] www.math.uci.edu/ ̃mfried/paplist-cov/UMStory.html. Algebraic equations in separated variables: (*) f(x)−g(y) = 0. Defines a projective nonsingular algebraic curve Xf,g with two projections to the (Riemann sphere) z-line Pz = C ∪ {∞}: f : Px → Pz and g : Py → Pz. Two problems from the 60s (Davenport’s and Shinzel’s) solved by the monodromy method . I explain that and the associated Genus ...

It has been known for a long time that the structure of a finite simple group is intimately connected with the structure of the centralizers of its involutions. An old result of Brauer and Fowler asserts, in fact, that there are at most a finite number of simple groups in which the centralizer of an involution has a given structure. A more specific, pioneering result of Brauer established that ...

A Beauville surface is a rigid complex surface of the form (C1 × C2)/G, where C1 and C2 are non-singular, projective, higher genus curves, and G is a finite group acting freely on the product. Bauer, Catanese, and Grunewald conjectured that every finite simple group G, with the exception of A5, gives rise to such a surface. We prove that this is so for almost all finite simple groups (i.e., wit...

My aim in this lecture will be to try to convince you that the classification of the finite simple groups is nearing its end. This is, of course, a presumptuous statement, since one does not normally announce theorems as "almost proved". But the classification of simple groups is unlike any other single theorem in the history of mathematics, since the final proof will cover at least 5,000 journ...