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Given a finite group
$G$
, define the minimal degree
$\unicode[STIX]{x1D707}(G)$
of
$G$
to be the least
$n$
such that
$G$
embeds into
$S_{n}$
. We call
$G$
exceptional if there is some
$N\unlhd G$
with
$\unicode[STIX]{x1D707}(G/N)>\unicode[STIX]{x1D707}(G)$
, in which case we call
$N$
distinguished. We prove here that a subgroup with no abelian composition factors is not distinguished.
Dolfi, Guralnick, Praeger and Spiga asked whether there exist infinitely many primitive groups of twisted wreath type with non-trivial coprime subdegrees. Here, we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with non-trivial coprime subdegrees. In particular, we define a primitive twisted wreath group G(m, q) constructed from the non-abelian simple group PSL(2, q) and a primitive permutation group of diagonal type with socle PSL(2, q)m, and determine many subdegrees for this group. A consequence is that we determine all values of m and q for which G(m, q) has non-trivial coprime subdegrees. In the case where m = 2 and
$q\notin \{7,11,29\}$
, we obtain a full classification of all pairs of non-trivial coprime subdegrees.
A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later, in 1921, he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this paper we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data.
Let
$K$
be a field that admits a cyclic Galois extension of degree
$n\geq 2$
. The symmetric group
$S_{n}$
acts on
$K^{n}$
by permutation of coordinates. Given a subgroup
$G$
of
$S_{n}$
and
$u\in K^{n}$
, let
$V_{G}(u)$
be the
$K$
-vector space spanned by the orbit of
$u$
under the action of
$G$
. In this paper we show that, for a special family of groups
$G$
of affine type, the dimension of
$V_{G}(u)$
can be computed via the greatest common divisor of certain polynomials in
$K[x]$
. We present some applications of our results to the cases
$K=\mathbb{Q}$
and
$K$
finite.
A generalisation of von Staudt’s theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.
We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.
For a finite group
$G$
, denote by
$\unicode[STIX]{x1D707}(G)$
the degree of a minimal permutation representation of
$G$
. We call
$G$
exceptional if there is a normal subgroup
$N\unlhd G$
with
$\unicode[STIX]{x1D707}(G/N)>\unicode[STIX]{x1D707}(G)$
. To complete the work of Easdown and Praeger [‘On minimal
faithful permutation representations of finite groups’, Bull. Aust.
Math. Soc.38(2) (1988), 207–220], for all primes
$p\geq 3$
, we describe an exceptional group of order
$p^{5}$
and prove that no exceptional group of order
$p^{4}$
exists.
We formulate some conjectures about the precise determination of the monodromy groups of certain rigid local systems on
$\mathbb{A}^{1}$
whose monodromy groups are known, by results of Kubert, to be finite. We prove some of them.
We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for the
$q$
-Schur superalgebras. This gives a combinatorialization of the relative norm approach developed in Du and Gu (A realization of the quantum supergroup
$\mathbf{U}(\mathfrak{g}\mathfrak{l}_{m|n})$
, J. Algebra 404 (2014), 60–99). We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for
$q$
-Schur superalgebras. We also construct infinitesimal and little
$q$
-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.
Given a finite group
$G$
, the generating graph
$\unicode[STIX]{x1D6E4}(G)$
of
$G$
has as vertices the (nontrivial) elements of
$G$
and two vertices are adjacent if and only if they are distinct and generate
$G$
as group elements. In this paper we investigate properties about the degrees of the vertices of
$\unicode[STIX]{x1D6E4}(G)$
when
$G$
is an alternating group or a symmetric group of degree
$n$
. In particular, we determine the vertices of
$\unicode[STIX]{x1D6E4}(G)$
having even degree and show that
$\unicode[STIX]{x1D6E4}(G)$
is Eulerian if and only if
$n\geqslant 3$
and
$n$
and
$n-1$
are not equal to a prime number congruent to 3 modulo 4.
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.
Let
$G$
be a finite group with
$\mathsf{soc}(G)=\text{A}_{c}$
for
$c\geq 5$
. A characterization of the subgroups with square-free index in
$G$
is given. Also, it is shown that a
$(G,2)$
-arc-transitive graph of square-free order is isomorphic to a complete graph, a complete bipartite graph with a matching deleted or one of
$11$
other graphs.
This article began as a study of the structure of infinite permutation groups
$G$
in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups
$G$
are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup
$M$
which is a divisible abelian
$p$
-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a
$p$
-adic vector space associated with
$M$
. This leads to our second variation, which is a study of the finite linear groups that can arise.
In this paper, a switch function-based gas-kinetic scheme (SF-GKS) is presented for the simulation of inviscid and viscous compressible flows. With the finite volume discretization, Euler and Navier-Stokes equations are solved and the SF-GKS is applied to evaluate the inviscid flux at cell interface. The viscous flux is obtained by the conventional smooth function approximation. Unlike the traditional gas-kinetic scheme in the calculation of inviscid flux such as Kinetic Flux Vector Splitting (KFVS), the numerical dissipation is controlled with a switch function in the present scheme. That is, the numerical dissipation is only introduced in the region around strong shock waves. As a consequence, the present SF-GKS can well capture strong shock waves and thin boundary layers simultaneously. The present SF-GKS is firstly validated by its application to the inviscid flow problems, including 1-D Euler shock tube, regular shock reflection and double Mach reflection. Then, SF-GKS is extended to solve viscous transonic and hypersonic flow problems. Good agreement between the present results and those in the literature verifies the accuracy and robustness of SF-GKS.
In this paper we describe methods for finding very small maximal subgroups of very large groups, with particular application to the subgroup 47:23 of the Baby Monster. This example is completely intractable by standard or naïve methods. The example of finding 31:15 inside the Thompson group
$\text{Th}$
is also discussed as a test case.
This paper contributes to the regular covers of a complete bipartite graph minus a matching, denoted
$K_{n,n}-nK_{2}$
, whose fiber-preserving automorphism group acts 2-arc-transitively. All such covers, when the covering transformation group
$K$
is either cyclic or
$\mathbb{Z}_{p}^{2}$
with
$p$
a prime, have been determined in Xu and Du [‘2-arc-transitive cyclic covers of
$K_{n,n}-nK_{2}$
’, J. Algebraic Combin.39 (2014), 883–902] and Xu et al. [‘2-arc-transitive regular covers of
$K_{n,n}-nK_{2}$
with the covering transformation group
$\mathbb{Z}_{p}^{2}$
’, Ars. Math. Contemp.10 (2016), 269–280]. Finally, this paper gives a classification of all such covers for
$K\cong \mathbb{Z}_{p}^{3}$
with
$p$
a prime.
For an odd prime
$p$
, a
$p$
-transposition group is a group generated by a set of involutions such that the product of any two has order 2 or
$p$
. We first classify a family of
$(G,2)$
-geodesic transitive Cayley graphs
${\rm\Gamma}:=\text{Cay}(T,S)$
where
$S$
is a set of involutions and
$T:\text{Inn}(T)\leq G\leq T:\text{Aut}(T,S)$
. In this case,
$T$
is either an elementary abelian 2-group or a
$p$
-transposition group. Then under the further assumption that
$G$
acts quasiprimitively on the vertex set of
${\rm\Gamma}$
, we prove that: (1) if
${\rm\Gamma}$
is not
$(G,2)$
-arc transitive, then this quasiprimitive action is the holomorph affine type; (2) if
$T$
is a
$p$
-transposition group and
$S$
is a conjugacy class, then
$p=3$
and
${\rm\Gamma}$
is
$(G,2)$
-arc transitive.
A graph is called arc-regular if its full automorphism group acts regularly on its arc set. In this paper, we completely determine all the arc-regular Frobenius metacirculants of prime valency.
Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math.310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group
$\text{A}_{39}$
. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group
$\text{A}_{40}$
.