A class of finite groups admitting certain sharp characters I

We consider the problem of determining the maximum possible out-degree d(n) of a digraph on n vertices which admits a sharply edge-transitive group. We show that d(n) > en/log log n for every n, while d(n) = in infinitely often. Also, d(n) = n -1 if and only if n is a prime power, whereas for non-prime-power values of n, we show that nd(n) tends to infinitely with n. The question has interesting grouptheoretic aspects. This and related problems generalise the existence question for projective planes.

Journal für die reine und angewandte Mathematik (Crelles Journal)

We construct sharply 2-transitive groups of characteristic 0 without regular normal subgroups. These groups act sharply 2-transitively by conjugation on their involutions. This answers a long-standing open question.

Journal of the London Mathematical Society, 1988

The concept of the 2-closure of a finite permutation group was introduced by Wielandt in [15]. We recall the definition: if G is a finite permutation group on a set ft, the 2-closure G <2) of G is the largest subgroup of Sym (Q) containing G which has the same orbits as G in the induced action on Q x Q. Wielandt obtains a number of properties of G <2> in [15], and there are some interesting applications in [12,13,15]. Clearly, if G is primitive of rank r on Q then so is G <2) , and G (2> is the largest subgroup of Sym (Q) containing G with this property. In particular, if G is 2-transitive on Q. then G (2> is the full symmetric group Sym (Q). It is natural to ask about the size of G (2) when G is simply primitive (that is, primitive but not 2-transitive). For nonabelian simple groups of order less than 10 6 this is one of the themes in [7]. In this paper we use the result of [9] to investigate G (2) whenever G is almost simple, that is, L o G < Aut L for some non-abelian simple group L. Our main result shows that if G is almost simple and simply primitive on ft, then G <2) is in general contained in the normalizer of L in Sym (Q). There are, however, some striking examples where this is not so. Apart from six 'sporadic' rank 3 examples which are probably well known, there are two infinite families with unbounded ranks. THEOREM 1. Let G be an almost simple group with L <i G ^ Aut L and L simple. Assume that G is simply primitive on a set ft of size n. Then G (2) is contained in the normalizer of L in Sym (Q) except in the following examples: (1) L = G 2 (q) with q^3,n = q\q z-\)/2 and the socle of G™ is Q 7 (#); (2) L = Cl 7 (q), n = ^3(^4-1)/(2, ?-1) and the socle of G™ is PQ£(tf); (3) six exceptional examples of rank 3 given in Table 1.

2013

In P G(2, q) a point set K is sharply transitive if the collineation group preserving K has a subgroup acting on K as a sharply transitive permutation group. By a result of Korchmáros, sharply transitive hyperovals only exist for a few values of q, namely q = 2, 4 and 16. In general, sharply transitive complete arcs of even size in P G(2, q) with q even seem to be sporadic. In this paper, we construct sharply transitive complete 6 ( √ q − 1)arcs for q = 4 2h+1, h ≤ 4. As far as we are concerned, these are the smallest known complete arcs in P G(2, 4 7) and in P G(2, 4 9); also, 42 seems to be a new value of the spectrum of the sizes of complete arcs in P G(2, 4 3). Our construction applies to any q which is an odd power of 4, but the problem of the completeness of the resulting sharply transitive arc remains open for q ≥ 4 11. In the second part of this paper, sharply transitive subsets arising as orbits under a Singer subgroup are considered and their characters, that is the possib...

Journal of Algebra, 1980

arXiv: Group Theory, 2016

We construct sharply 2-transitive groups of characteristic 0 without non-trivial abelian normal subgroups. These groups act sharply 2-transitively by conjugation on their involutions. This answers a longstanding open question.

Proceedings of the American Mathematical Society, 2021

We give examples of countable linear groups Γ < SL3(R), with no nontrivial normal abelian subgroups, that admit a faithful sharply 2-transitive action on a set. Without the linearity assumption, such groups were recently constructed by Rips, Segev, and Tent in [RST17]. Our examples are of permutational characteristic 2, in the sense that involutions do not fix a point in the 2-transitive action.

2014

A linear group G on a finite vector space V, (that is, a subgroup of GL(V)) is called (1/2)-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the (1/2)-transitive linear groups. As a consequence we complete the determination of the finite (3/2)-transitive permutation groups -- the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the finite (k+1/2)-transitive permutation groups for integers k > 1.

Communications in Algebra, 2019

We mainly introduce finitary series of groups which can be represented as permutation or linear groups. Also we define (finitary permutation)-hypercentral (nilpotent) and (finitary linear)-hypercentral (nilpotent) groups. We give certain applications of the introduced concepts to perfect locally finite barely transitive p-groups.

Discrete Mathematics, 2010

, the codewords of small weight in the dual code of the code of points and lines of Q(4, q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q + (5, q) and H(5, q 2 ), and we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces of the classical polar spaces. Furthermore, we investigate the codewords with the largest weights in these codes, where for q even and k sufficiently small, we determine the maximum weight and characterise the codewords of maximum weight. Moreover, we show that there exists an interval such that for every even number w in this interval, there is a codeword in the dual code of Q + (5, q), q even, with weight w and we show that there is an empty interval in the weight distribution of the dual of the code of Q(4, q), q even. To prove this, we show that a blocking set of Q(4, q), q even, of size q 2 + 1 + r, where 0 < r < (q + 4)/6, contains an ovoid of Q(4, q), improving on [5, Theorem 9].