We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T of a maximal subgroup of T not an alternating group we prove that, with finitely many ...exceptions, the maximum element order is at most m(T). These results are applied to determine all primitive permutation groups on a set of size n.
For a positive integer
k
, a group
G
is said to be totally
k
-closed if for each set
Ω
upon which
G
acts faithfully,
G
is the largest subgroup of
Sym
(
Ω
)
that leaves invariant each of the
G
-orbits ...in the induced action on
Ω
×
⋯
×
Ω
=
Ω
k
. Each finite group
G
is totally |
G
|-closed, and
k
(
G
) denotes the least integer
k
such that
G
is totally
k
-closed. We address the question of determining the closure number
k
(
G
) for finite simple groups
G
. Prior to our work it was known that
k
(
G
)
=
2
for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that
k
(
G
)
⩾
3
for all other finite simple groups. We determine the value for the alternating groups, namely
k
(
A
n
)
=
n
-
1
. In addition, for all simple groups
G
, other than alternating groups and classical groups, we show that
k
(
G
)
⩽
7
. Finally, if
G
is a finite simple classical group with natural module of dimension
n
, we show that
k
(
G
)
⩽
n
+
2
if
n
⩾
14
, and
k
(
G
)
⩽
⌊
n
/
3
+
12
⌋
otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on
n
and the type of
G
) on the base sizes of the primitive actions of
G
, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups.
On flag‐transitive imprimitive 2‐designs Devillers, Alice; Praeger, Cheryl E.
Journal of combinatorial designs,
July 2021, 2021-07-00, 20210701, Letnik:
29, Številka:
8
Journal Article
Recenzirano
Odprti dostop
In 1987, Huw Davies proved that, for a flag‐transitive point‐imprimitive 2‐
(
v
,
k
,
λ
) design, both the block‐size
k and the number
v of points are bounded by functions of
λ, but he did not make ...these bounds explicit. In this paper we derive explicit polynomial functions of
λ bounding
k and
v. For
λ
⩽
4 we obtain a list of “numerically feasible” parameter sets
v
,
k
,
λ together with the number of parts and part‐size of an invariant point‐partition and the size of a nontrivial block‐part intersection. Moreover from these parameter sets we determine all examples with fewer than 100 points. There are exactly 11 such examples, and for one of these designs, a flag‐regular, point‐imprimitive
2
−
(
36
,
8
,
4
) design with automorphism group
S
6, there seems to be no construction previously available in the literature.
Let V be a d-dimensional vector space over a finite field F equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose |F|=q2 if V is hermitian, and |F|=q otherwise. Given ...integers e,e′ such that e+e′⩽d, we estimate the proportion of pairs (U,U′), where U is a non-degenerate e-subspace of V and U′ is a non-degenerate e′-subspace of V, such that U∩U′=0 and U⊕U′ is non-degenerate (the sum U⊕U′ is direct and usually not perpendicular). The proportion is shown to be positive and at least 1−c/q>0 for some constant c. For example, c=7/4 suffices in both the unitary and symplectic cases. The arguments in the orthogonal case are delicate and assume that dim(U) and dim(U′) are even, an assumption relevant for an algorithmic application (which we discuss) for recognising finite classical groups. We also describe how recognising a classical groups G relies on a connection between certain pairs (U,U′) of non-degenerate subspaces and certain pairs (g,g′)∈G2 of group elements where U=im(g−1) and U′=im(g′−1).
Generating Infinite Digraphs by Derangements Horsley, Daniel; Iradmusa, Moharram; Praeger, Cheryl E
Quarterly journal of mathematics,
09/2021, Letnik:
72, Številka:
3
Journal Article
Recenzirano
Odprti dostop
Abstract
A set $\mathcal{S}$ of derangements (fixed-point-free permutations) of a set V generates a digraph with vertex set V and arcs $(x,x^{\,\sigma})$ for x ∈ V and $\sigma\in\mathcal{S}$. We ...address the problem of characterizing those infinite (simple loopless) digraphs which are generated by finite sets of derangements. The case of finite digraphs was addressed in an earlier work by the second and third authors. A criterion is given for derangement generation which resembles the criterion given by De Bruijn and Erdős for vertex colourings of graphs in that the property for an infinite digraph is determined by properties of its finite sub-digraphs. The derangement generation property for a digraph is linked with the existence of a finite 1-factor cover for an associated bipartite (undirected) graph.
The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive ...generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group
$G$
preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on
$G$
, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that
$G$
cannot have holomorph compound O’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.
Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-imprimitive, block-transitive 2-design. To do this they introduced two integer parameters
m
,
n
, now ...called Delandtsheer–Doyen parameters, linking the block size with the parameters of an associated imprimitivity system on points. We show that the Delandtsheer–Doyen parameters provide upper bounds on the permutation ranks of the groups induced on the imprimitivity system and on a class of the system. We explore extreme cases where these bounds are attained, give a new construction for a family of designs achieving these bounds, and pose several open questions concerning the Delandtsheer–Doyen parameters.
We present a new proof, which is independent of the finite simple group classification and applies also to infinite groups, that quasiprimitive permutation groups of simple diagonal type cannot be ...embedded into wreath products in product action. The proof uses several deep results that concern factorisations of direct products involving subdirect subgroups. We find that such factorisations are controlled by the existence of uniform automorphisms.
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