On cyclic semi-regular subgroups of certain 2-transitive permutation groups (original) (raw)
Related papers
Two Theorems on Doubly Transitive Permutation Groups
Journal of the London Mathematical Society, 1973
In a series of papers [3, 4 and 5] on insoluble (transitive) permutation groups of degree p = 2q +1, where p and q are primes, N. Ito has shown that, apart from a small number of exceptions, such a group must be at least quadruply transitive. One of the results which he uses is that an insoluble group of degree p = 2q +1 which is not doubly primitive must be isomorphic to PSL (3, 2) with p = 7. This result is due to H. Wielandt, and ltd gives a proof in [3]. It is quite easy to extend this proof to give the following result: a doubly transitive group of degree 2q + l, where q is prime, which is not doubly primitive, is either sharply doubly transitive or a group of automorphisms of a block design with A = 1 and k = 3. Our notation for the parameters of a block design, v, b, k, r, X, is standard; see [9]. In this paper we shall prove two results about doubly transitive but not doubly primitive groups which resemble the two results mentioned above.
Transitive Subgroups of Primitive Permutation Groups
Journal of Algebra, 2000
to helmut wielandt on the occasion of his 90th birthday We investigate the finite primitive permutation groups G which have a transitive subgroup containing no nontrivial subnormal subgroup of G. The conclusion is that such primitive groups are rather rare, and that their existence is intimately connected with factorisations of almost simple groups. A corollary is obtained on primitive groups which contain a regular subgroup. Heavily involved in our proofs are some new results on subgroups of simple groups which have orders divisible by various primes. For example, another corollary implies that for every simple group T apart from L 3 3 , U 3 3 , and L 2 p with p a Mersenne prime, there is a collection consisting of two or three odd prime divisors of T , such that if M is a subgroup of T of order divisible by every prime in , then M is divisible by all the prime divisors of T , and we obtain a classification of such subgroups M.
On the Sylow subgroups of a doubly transitive permutation group III
Bulletin of the Australian Mathematical Society, 1975
Let G be a 2-transitive permutation group of a set Ω of n points and let P be a Sylow p-subgroup of G where p is a prime dividing |G|. If we restrict the lengths of the orbits of P, can we correspondingly restrict the order of P? In the previous two papers of this series we were concerned with the case in which all P–orbits have length at most p; in the second paper we looked at Sylow p–subgroups of a two point stabiliser. We showed that either P had order p, or G ≥ An, G = PSL(2, 5) with p = 2, or G = M11 of degree 12 with p = 3. In this paper we assume that P has a subgroup Q of index p and all orbits of Q have length at most p. We conclude that either P has order at most p2, or the groups are known; namely PSL(3, p) ≤ G ≤ PGL(3, p), ASL(2, p) ≤ G ≤ AGL(2, p), G = PΓL,(2, 8) with p = 3, G = M12 with p = 3, G = PGL(2, 5) with p = 2, or G ≥ An with 3p ≤ n < 2p2; all in their natural representations.
The classification of (3/2)-transitive permutation groups and (1/2)-transitive linear groups
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.
Proceedings of the American Mathematical Society
A linear group G ≤ G L ( V ) G\le GL(V) , where V V is a finite vector space, is called 1 2 \frac {1}{2} -transitive if all the G G -orbits on the set of nonzero vectors have the same size. We complete the classification of all the 1 2 \frac {1}{2} -transitive linear groups. As a consequence we complete the determination of the finite 3 2 \frac {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 ( k + 1 2 ) (k+\frac {1}{2}) -transitive groups for integers k ≥ 2 k\ge 2 .
Sylow subgroups of transitive permutation groups II
Journal of the Australian Mathematical Society, 1977
Let G be a transitive permutation group on a finite set of n points, and let P be a Sylow p-subgroup of G for some prime p dividing |G|. We are concerned with finding a bound for the number f of points of the set fixed by P. Of all the orbits of P of length greater than one, suppose that the ones of minimal length have length q, and suppose that there are k orbits of P of length q. We show that f ≦ kp − ip(n), where ip(n) is the integer satisfying 1 ≦ ip(n) ≦ p and n + ip(n) ≡ 0(mod p). This is a generalisation of a bound found by Marcel Herzog and the author, and this new bound is better whenever P has an orbit of length greater than the minimal length q.
Transitive Permutation Groups Without Semiregular Subgroups
Journal of the London Mathematical Society, 2002
A transitive finite permutation group is called elusive if it contains no nontrivial semiregular subgroup. The purpose of the paper is to collect known information about elusive groups. The main results are recursive constructions of elusive permutation groups, using various product operations and affine group constructions. A brief historical introduction and a survey of known elusive groups are also included. In a sequel, Giudici has determined all the quasiprimitive elusive groups.
On Transitive Permutation Groups with Primitive Subconstituents
Bulletin of the London Mathematical Society, 1999
Let G be a transitive permutation group on a set Ω such that, for ω ? Ω, the stabiliser G ω induces on each of its orbits in ΩBoωq a primitive permutation group (possibly of degree 1). Let N be the normal closure of G ω in G. Then (Theorem 1) either N factorises as N l G ω G δ for some ω, δ ? Ω, or all unfaithful G ω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the G ω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω.
Doubly transitive permutation groups which are not doubly primitive
Journal of Algebra, 1978
Some results on doubly transitive but not doubly primitive permutation groups are proved, giving more evidence to Atkinson's conjecture [3]. Among other results, we characterize the group S?(q) as a group satisfying the condition of the title and prove some sufficient conditions for such a group to be an automorphism group of a nontrivial block design with h = 1. Recently there has been considerable interest in the structure'of the one point stabilizer of a doubly transitive permutation group G on a set Q. One problem is to describe those doubly transitive groups for which G, , 01 E Q, is imprimitive on Q-{a!}. Our general assumption is: Hypothesis (A): G is a doubly transitive permutation group on a set Q.
Journal of Algebra, 1977
Suppose that G is a doubly transitive permutation group on a finite set Q and that for o! in 52 the stabilizer G, has a set C = {B, ,..., B,} of nontrivial blocks of of imprimitivity in 52-{a}, that is, 1 Bi > 1 and / C i > 1. In two previous papers [5, 61 it was shown that apart from a few known groups, the setwise stabilizer of B, in G, acts faithfully on B, if G," is the alternating or symmetric group, one of the Mathieu groups, or a normal extension of PSL(2,q) in their usual representations. The raises the question: QUESTION. If G," is multiply transitive, is it possible to characterize the groups G for which the setwise stablizer in G, of the block B of 2 does not act faithfully on B ? The only groups I know of in which G, is 2-transitive on z1 and for B in Z, the stablizer of B is not faithful on B, are the following: (i) PSL(n, Q) < G < PrL(n, q), for n > 3 in its natural representation. If B E Z then B u (a> is a line and PSL(n-I, q)-< G," < PTL(n-1, 4). (ii) PSU(3, q) < G < PlYJ(3, q) p ermuting the set of absolute points of the projective plane over a field of q2 elements. If B E 1: then B u {a} is a nonabsolute line, 1 B 1 = q, 1 Z 1 = q2, and G,z has a regular normal subgroup of order q?. (iii) G = A, acting on the 15 points of the projective geometry of dimension 3 over a field of two elements; G, N PSL(2, 7) acts 2-transitively on the set of lines containing 01. (iv) G has a regular normal subgroup. If we drop the assumption that G," is 2-transitive, then we have another family of examples. (v) A group G of Ree type R(q) is 2-transitive on q3 + 1 points. For any two distinct points 01, p, there is a unique nontrivial element g in Go,* which 433