Transitive Permutation Groups Of Degree Research Articles

The Monodromy Group of pq-Covers

In this work we study the monodromy group of covers $\varphi \circ \psi$ of curves \linebreak $\mathcal{Y}\xrightarrow {\quad {\psi}} \mathcal{X} \xrightarrow {\quad \varphi} \mathbb{P}^{1}$, where $\psi$ is a $q$-fold cyclic \'etale cover and $\varphi$ is a totally ramified $p$-fold cover, with $p$ and $q$ different prime numbers with $p$ odd. We show that the Galois group $\mathcal{G}$ of the Galois closure $\mathcal{Z}$ of $\varphi \circ \psi$ is of the form $ \mathcal{G} = \mathbb{Z}_q^s \rtimes \mathcal{U}$, where $0 \leq s \leq p-1$ and $\mathcal{U}$ is a simple transitive permutation group of degree $p$. Since the simple transitive permutation group of prime degree $p$ are known, and we construct examples of such covers with these Galois groups, the result is very different from the previously known case when the cover $\varphi$ was assumed to be cyclic, in which case the Galois group is of the form $ \mathcal{G} = \mathbb{Z}_q^s \rtimes \mathbb{Z}_p$. Furthermore, we are able to characterize the subgroups $\mathcal{H}$ and $\mathcal{N}$ of $\mathcal{G}$ such that $\mathcal{Y} = \mathcal{Z}/\mathcal{N}$ and $X = \mathcal{Z}/\mathcal{H}$.

Semiregular elements in transitive 2-closed permutation groups of certain degrees

A nonidentity element of a permutation group is said to be semiregular provided all of its cycles in its cycle decomposition are of the same length. It is known that semiregular elements exist in transitive 2-closed permutation groups of square-free degree and in some special cases when the degree is divisible by a square of a prime. In this paper it is shown that semiregular elements exist in transitive 2-closed permutation groups of the following degrees 16p, where is a prime, where is a prime, 12pq, where are primes, and either or 18pq, where are primes and where are primes, and or qr < s, and 4pqrs, where are primes, pqr < s, and As a corollary, a 2-closed transitive permutation group of degree and different from 72 and 96 contains semiregular elements.

Three local actions in 6‐valent arc‐transitive graphs

AbstractIt is known that there are precisely three transitive permutation groups of degree 6 that admit an invariant partition with three parts of size 2 such that the kernel of the action on the parts has order 4; these groups are called , and . For each , we construct an infinite family of finite connected 6‐valent graphs and arc‐transitive groups such that the permutation group induced by the action of the vertex‐stabiliser on the neighbourhood of a vertex is permutation isomorphic to , and such that is exponential in . These three groups were the only transitive permutation groups of degree at most 7 for which the existence of such a family was undecided. In the process, we construct an infinite family of cubic 2‐arc‐transitive graphs such that the dimension of the 1‐eigenspace over the field of order 2 of the adjacency matrix of the graph grows linearly with the order of the graph.

On the minimal degree of a transitive permutation group with stabilizer a 2-group

Abstract The minimal degree of a permutation group G is defined as the minimal number of non-fixed points of a non-trivial element of G. In this paper, we show that if G is a transitive permutation group of degree n having no non-trivial normal 2-subgroups such that the stabilizer of a point is a 2-group, then the minimal degree of G is at least 2 3 ⁢ n {\frac{2}{3}n} . The proof depends on the classification of finite simple groups.

A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

Abstract We show that there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by a | G : H | 3 / 2 {a\lvert G:H\rvert^{3/2}} . In particular, a transitive permutation group of degree n has at most a ⁢ n 3 / 2 {an^{3/2}} maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stronger bound, that is, the number of maximal subgroups of G containing H is at most | G : H | - 1 {\lvert G:H\rvert-1} .

Some generalized characters associated to a transitive permutation group

Abstract Let G be a finite transitive permutation group of degree n, with point stabilizer H ≠ 1 {H\neq 1} and permutation character π. For every positive integer t, we consider the generalized character ψ t = ρ G - t ⁢ ( π - 1 G ) {\psi_{t}=\rho_{G}-t(\pi-1_{G})} , where ρ G {\rho_{G}} is the regular character of G and 1 G {1_{G}} the 1-character. We give necessary and sufficient conditions on t (and G) which guarantee that ψ t {\psi_{t}} is a character of G. A necessary condition is that t ≤ min ⁡ { n - 1 , | H | } {t\leq\min\{n-1,\lvert H\rvert\}} , and it turns out that ψ t {\psi_{t}} is a character of G for t = n - 1 {t=n-1} resp. t = | H | {t=\lvert H\rvert} precisely when G is 2-transitive resp. a Frobenius group.

Comparing the expected number of random elements from the symmetric and the alternating groups needed to generate a transitive subgroup

Given a transitive permutation group of degree n , we denote by e T ( G ) the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators of a transitive subgroup of G is found. We compare e T (Sym( n )) and e T (Alt( n )) .

An extension of the classification of high rank regular polytopes

Up to isomorphism and duality, there are exactly two non-degenerate abstract regular polytopes of rank greater than $n-3$, one of rank $n-1$ and one of rank $n-2$, with automorphism groups that are transitive permutation groups of degree $n\geq 7$. In this paper we extend this classification of high rank regular polytopes to include the ranks $n-3$ and $n-4$. The result is, up to a isomorphism and duality, seven abstract regular polytopes of rank $n-3$ for each $n\geq 9$, and nine abstract regular polytopes of rank $n-4$ for each $n \geq 11$. Moreover we show that if a transitive permutation group $\Gamma$ of degree $n \geq 11$ is the automorphism group of an abstract regular polytope of rank at least $n-4$, then $\Gamma\cong S_n$.

Locally elusive classical groups

Let G be a transitive permutation group of degree n with point stabiliser H and let r be a prime divisor of n. We say that G is r-elusive if it does not contain a derangement of order r. The problem of determining the r-elusive primitive groups can be reduced to the almost simple case, and the purpose of this paper is to complete the study of r-elusivity for almost simple classical groups. Building on our earlier work for geometric actions of classical groups, in this paper we handle the remaining non-geometric actions where H is almost simple and irreducible. This requires a completely different approach, using tools from the representation theory of quasisimple groups.

Permutation groups and derangements of odd prime order

Let G be a transitive permutation group of degree n. We say that G is 2′-elusive if n is divisible by an odd prime, but G does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive 2′-elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate automorphisms of finite arc-transitive graphs of prime valency.

Generating minimally transitive permutation groups

We prove that each minimally transitive permutation group of degree n can be generated by μ(n)+1 elements, where μ(n):=max⁡{m:there exists a prime p such that pm divides n}.

On the diameter of permutation groups

Given a finite group G and a set A of generators, the diameter diam(G(G,A)) of the Cayley graph G(G,A) is the smallest l such that every element of G can be expressed as a word of length at most l in A?A -1 . We are concerned with bounding diam(G):=max A diam(G(G,A)) . It has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n , but the best previously known upper bound was exponential in nlogn - - - - - v . We give a quasipolynomial upper bound, namely, diam(G)=exp(O((logn) 4 loglogn))=exp((loglog|G|) O(1) ) for G=Sym(n) or G=Alt(n) , where the implied constants are absolute. This addresses a key open case of Babai�s conjecture on diameters of simple groups. By a result of Babai and Seress (1992), our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n .

On the degree of compositum of two number fields

AbstractWe prove that the sum of two algebraic numbers both of degree 6 cannot be of degree 8. The triplet (6, 6, 8) was the only undecided case in the previous characterization of all positive integer triplets (a, b, c), with a ≤ b ≤ c and b ≤ 6, for which there exist algebraic numbers α, β and γ of degrees a, b and c, respectively, such that α + β + γ = 0. Now, this characterization is extended up to b ≤ 7. We also solve a similar problem for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(a,b,c) \in \mathbb N^3$\end{document} with a ≤ b ≤ 7 by finding for which positive integers a, b, c there exist number fields of degrees a and b such that their compositum has degree c. We show that the problem of describing all compositum‐feasible triplets can be reduced to some finite computation in transitive permutation groups of degree c which occur as Galois groups of irreducible polynomials of degree c. In particular, by exploiting the properties of the projective special linear group PSL(2, 7) of order 168, we prove that the triplet (7, 7, 28) is compositum‐feasible. In the opposite direction, the triplet (p, p, p(p − ℓ)), where ℓ ≥ 2 is an integer and p &gt; ℓ2 − ℓ + 1 is a prime number, is shown to be not compositum‐feasible, not sum‐feasible and not product‐feasible.

Metacirculant tournaments whose order is a product of two distinct primes

In this paper, we prove that non-circulant vertex-transitive tournaments of order p q , where p and q are distinct odd primes, are metacirculant tournaments (defined in Definition 2.1) satisfying some special conditions; see Theorem 1.2. So, in combination with the work in Jing Xu (2010) [11], a complete classification of vertex-transitive p q -tournaments is obtained. As a by-product, we construct examples of non-Cayley vertex-transitive p q -tournaments where q 2 | ( p − 1 ) in Example 2.5. Moreover, applying the classification of vertex-transitive p q -tournaments, we determine all 2-closed (in Wielandt’s sense) odd-order transitive permutation groups of degree p q and show that each of them is the full automorphism group of some tournament.

Vertex-transitive tournaments of order a product of two distinct primes

In this paper, we begin a classification of the vertex-transitive tournaments of order pq where p and q are distinct odd primes. In particular we characterize the pq -circulant tournaments; see Theorem 4.3. Moreover, we determine 2-closed (in Wielandt's sense) odd-order transitive permutation groups of degree p and pq by using the classifications of vertex-transitive tournaments of order p and pq . We prove that each such 2-closed group containing a cyclic regular subgroup is the full automorphism group of some circulant tournament.

General form of Humbert's modular equation for curves with real multiplication of $\Delta =5$

We study Humbert's modular equation which characterizes curves of genus two having real multiplication by the quadratic order of discriminant 5. We give it a simple, but general expression as a polynomial in $x_1,\ldots ,x_6$ the coordinate of the Weierstrass points, and show that it is invariant under a transitive permutation group of degree 6 isomorphic to $\frak S_5$. We also prove the rationality of the hypersurface in P5 defined by the generalized modular equation.

A note on transitive permutation groups of degree twice a prime

In this note, we consider transitive permutation groups of degree 2 p , where p is an odd prime, that admit blocks of imprimitivity of size 2 but no blocks of imprimitivity of size p .

Transitive Permutation Groups with Cyclic Point Stabilizers of Maximum Order

We prove that a transitive permutation group of degree n with a cyclic point stabilizer and whose order is n(n-1) is isomorphic to the affine group of degree 1 over a field with n elements. More generally we show that if a finite group G has an abelian and core-free Hall subgroup Q, then either Q has a small order (2|Q|2 < |G|) or G is a direct product of 2-transitive Frobenius groups.

Abelian point stabilizers in transitive permutation groups

In this note we prove that if the point stabilizer $A$ in a transitive permutation group of degree $m$ is abelian, then the exponent of $A$ is less than $m$. This extends an earlier result of Andrea Lucchini, who proved this in the case where $A$ is cyclic.

Asymptotic Results for Transitive Permutation Groups

In this paper we give answers to some open questions concerning generation and enumeration of finite transitive permutation groups. In [1], Bryant, Kovács and Robinson proved that there is a number c′ such that each soluble transitive permutation group of degree n⩾2 can be generated by [ c ′ n / l o g n ] elements, and later A. Lucchini [5] extended this result (with a different constant c′) to finite permutation groups containing a soluble transitive subgroup. We are now able to prove this theorem in full generality, and this solves the question of bounding the number of generators of a finite transitive permutation group in terms of its degree. The result obtained is the following. 1991 Mathematics Subject Classification 20B05, 20D60.