Showing papers in "Archiv der Mathematik in 1984"

On minimal degrees and base sizes of primitive permutation groups

Abstract: The minimal degree/~ (G) of a primitive permutat ion group G of degree n on a set ~, that is, the smallest number of points moved by any non-identity element of G, has been the subject of considerable study in the classical theory of permutat ion groups (see w 15 of [9], for example). The best result available before the completion of the classification of finite simple groups was one due to Babai (see Theorem 6.14 of [1]): provided that A, :g G, we have #(G) > 89 1). In this note we use the classification theorem to improve this result. This is done by consideration of bases of G: a subset A of O is a base of G if the pointwise stabiliser of A in G is the identity (A is called afixing set in [1]). Denote by b (G) the minimum size of any base of G. It is shown in [1] that if G is simply primitive then b (G) < 4 x /~ log n. Using the classification theorem we prove the following result.

Modules without self-extensions and Nakayama's conjecture

Abstract: Let A be a self-injective artin algebra and X a finitely generated module. In this paper, we are mainly concerned with the problem posed by Tachikawa [16]: I f Ext"(X, X) = 0 for all n >_ 1, is then X projective? This problem arose from Nakayama's conjecture [11] which states that a finite dimensional algebra of infinite dominant dimension is selfinjective. It should be noted that if the answer to the above problem is affirmative, then Nakayama's conjecture is true for finite dimensional QF-3 algebras R with minimal faithful ideal Re such that e R e is self-injective (see [16] for details). Note also that a finite dimensional algebra of dominant dimension > 1 is QF-3 [15]. Tachikawa [16] showed that the above problem has affirmative answer in case A is a group algebra of a p-group, and this result was recently generalized by Schulz [14] to the case of A being a group algebra of an arbitrary finite group. We will also give several partial answers to the above problem and thus to Nakayama's conjecture. Throughout this paper, all modules are finitely generated modules over artin algebras, and most modules are fight modules. Given an artin algebra A over the center C, we denote by D the duality H o m e ( , / ) , where I is the injective envelope of C/tad C over C, and by Jf~a the category of the finitely generated fight A-modules. For modules X, Y we denote by Horn(X, Y) the factor group of Horn(X, Y) modulo the subgroup of the homomorphisms which factor through projective modules, and for a module X we denote by f2"X the n-th syzygy module of X. Note that in case A is self-injective, induces a self-equivalence of the stable category [I0]. Moreover, in case A is symmetric, we have (22 = D Tr. We refer to [1] and [2] for Auslander-Reiten sequences, irreducible homomorphisms and so on, and to [13] and [8] for (stable) Auslander-Reiten quivers. In what follows, D Tr and Tr D are denoted by z and ~1 respectively, and the "AuslanderReiten sequence" and the "Auslander-Reiten quiver" are siflaply written by the "ARsequence" and the "AR-quiver" respectively.

On groups with no regular orbits on the set of subsets

Abstract: Let G be a permutation group on a finite set f2 of size n. Then G acts naturally on the set P (f2) of all subsets of f2. In this note we shall show that if G is primitive on f2 and A, $ G then in all except finitely many cases G has a regular orbit on P (O). We were led to consider this question by some recent work of Siemons and Wagner [5] and Inglis [4], where the following situation was considered. Given permutation groups G and H on the same finite set f2, write G ~ H if G and H have the same orbits on P (f2). It is shown in [4] and [5] that if G is primitive, A, ~ G and G ~ H then with a few explicitly known exceptions the same primes divide [GI and [HI. It is an immediate consequence of our result that under the same hypotheses with finitely many exceptions in fact G = H. (Indeed, with the weaker assumption that the longest orbits of G and H on P (f2) have the same length, we can conclude that [ G I = I HI with only finitely many exceptions.) Unlike the arguments in [4] and [5], ours require the classification of finite simple groups. We do however get a good deal of information even without the classification and can obtain our conclusion by elementary methods if G is 2-transitive on f2. The problem of determining pairs G and H of groups with G ~ H without the assumption of primitivity seems more difficult. Note, however, that if G ~ H and G is primitive then so is H: for G is primitive if and only if the only orbits of G on P (O) whose members partition f2 are the trivial ones. Similarly, but more easily, if G ~ H and G is transitive then so is H. We start with a preliminary well-known result. Recall that the minimal degree m of a permutation group G is the least number of points moved by a non-identity element of G, while the base size b is the least number of points of which the stabilizer is the identity.

Nonabelian crowns and Schunck classes of finite groups

Abstract: 1. Some facts about primitive groups. A group G is said to be primitive ff it has a primitive and faithful permutation representation, i.e. if G has a maximal subgroup U such that core~U ---1. By Baer (w 2 of [1]), if P denotes the class of all primitive groups, one has (we use the notation in [4]): P = P1 0 P2 o P3 where G e P1 if and only ifGisprimitive and has a (single) soluble minimal normal subgroup, G e P2 if and only if G is primitive and has a (unique) minimal normal subgroup R such that Ca(R) ---1, and G e P3 if and only ifG is primitive and has precisely two distinct (nonsoluble) minimal normal subgroups C and D such that Ca(C) = D and Ca(D) = C. A primitive group is said to be of type i e (1, 2, 3} if it belongs to Pi.

An embedding theorem for profinite groups

Abstract: A well-known result of Higman, Neumann and Neumann [1] asserts that every countable group can be embedded in a 2-generator group. In this note we record a similar but stronger result for profinite groups. By a 2-generator profinite group we mean a profinite group which can be generated as a topological group by two of its elements; of course any closed subgroup of such a group must be countably based. Our result is the following:

A note on a separation problem

Abstract: The author proves the following theorem: Let A0 be a closed 1-dimensional semianalytic germ at the origin 0∈Rn. Let Z be a semianalytic set in Rn whose germ Z0 at 0 is closed and A0∩Z0={0}. Then there exists a polynomial h∈R[x1,⋯,xn] such that h∣Z∖{0}>0 and h∣A0∖{0}<0. The proof is by induction on the number of blowing-ups needed to "solve" the set A0. Some implications are then given, in particular a similar result for semialgebraic sets in Rn and polynomials.