Let $m$ be a positive integer and let $Ω$ be a finite set. The $m$-closure of $G\leq\operatorname{Sym}(Ω)$ is the largest permutation group on $Ω$ having the same orbits as $G$ in its induced action on the Cartesian product $Ω^m$. The $1$-closure and $2$-closure of a solvable permutation group need not be solvable. We prove that the $m$-closure of a solvable permutation group is always solvable for $m\geq3$.
Suppose that $G$ is a finite, transitive, solvable permutation group acting on a set $S$ with $n$ elements. Let $G_0$ be the stabilizer of a point $α\in Ω$. Define the rank of a permutation group, denoted $r(G),$ as the number of distinct orbits of $G_0$ in $S$ (including the trivial orbit $\{α\}$). Huppert \cite{Huppert} and Foulser \cite{Foulser} classified all finite, solvable, permutation groups of rank two and three respectively, and Foulser restricted the rank four groups to a small list of possibilities. This paper completes the classification of all groups of rank less than $5$ by expl…
Let $G$ be a permutation group on a set $Ω$, and $k$ a positive integer. The $k$-closure $G^{(k)}$ of $G$ is the largest subgroup of $\operatorname{Sym}(Ω)$, with the same as $G$ orbits of componentwise action on $Ω^k$. We prove that the $k$-closure of a finite nilpotent permutation group is the direct product of $k$-closures of its Sylow subgroups.
The $2$-closure $\overline{G}$ of a permutation group $G$ on $Ω$ is defined to be the largest permutation group on $Ω$, having the same orbits on $Ω\timesΩ$ as $G$. It is proved that if $G$ is supersolvable, then $\overline{G}$ can be found in polynomial time in $|Ω|$. As a byproduct of our technique, it is shown that the composition factors of $\overline{G}$ are cyclic or alternating of prime degree.
A permutation group $G$ on $Ω$ is called a rank 3 group if it has precisely three orbits in its induced action on $Ω\times Ω$. The largest permutation group on $Ω$ having the same orbits as $G$ on $Ω\times Ω$ is called the 2-closure of $G$. A description of 2-closures of rank 3 groups is given. As a special case, it is proved that 2-closure of a primitive one-dimensional affine rank 3 permutation group of sufficiently large degree is also affine and one-dimensional.
A finite permutation group $G$ on $Ω$ is called a rank 3 group if it has precisely three orbits in its induced action on $Ω\times Ω$. The largest permutation group on $Ω$ having the same orbits as $G$ on $Ω\times Ω$ is called the 2-closure of $G$. We construct a polynomial-time algorithm which given generators of a rank 3 group computes generators of its 2-closure.
Let $G$ be a finite almost simple group with socle $G_0$. A (nontrivial) factorization of $G$ is an expression of the form $G=HK$, where the factors $H$ and $K$ are core-free subgroups. There is an extensive literature on factorizations of almost simple groups, with important applications in permutation group theory and algebraic graph theory. In a recent paper, Li and Xia describe the factorizations of almost simple groups with a solvable factor $H$. Several infinite families arise in the context of classical groups and in each case a solvable subgroup of $G_0$ containing $H \cap G_0$ is iden…
Let $m$ be a positive integer and let $Ω$ be a finite set. The $m$-closure of $G\le$Sym$(Ω)$ is the largest permutation group on $Ω$ having the same orbits as $G$ in its induced action on the Cartesian product $Ω^m$. The exact formula for the $m$-closure of the wreath product in product action is given. As a corollary, a sufficient condition is obtained for this $m$-closure to be included in the wreath product of the $m$-closures of the factors.
School of Mathematics and Statistics, Guangxi Normal University（广西师范大学数学与统计学院）；Department of Mathematics, Shanghai University（上海大学数学系）；Newtouch Center for Mathematics of Shanghai University（上海大学新拓数学中心）
Let $G$ be a permutation group on a finite set $Ω$. The $k$-closure $G^{(k)}$ of the group $G$ is the largest subgroup of $\operatorname{Sym}(Ω)$ having the same orbits as $G$ on the $k$-th Cartesian power $Ω^k$ of $Ω$. A group $G$ is called $\frac{3}{2}$-transitive if its transitive and the orbits of a point stabilizer $G_α$ on the set $Ω\setminus\{α\}$ are of the same size greater than one. We prove that the $2$-closure $G^{(2)}$ of a $\frac{3}{2}$-transitive permutation group $G$ can be found in polynomial time in size of $Ω$. In addition, if the group $G$ is not $2$-transitive, then for ev…
Let $N$ be a normal subgroup of a finite group $G$. For a faithful $N$-set $Δ$, applying the university embedding theorem one can construct a faithful $G$-set $Ω$. In this short note, it is proved that if the $2$-closure of $N$ in $Ω$ is equal to $N$, then the $2$-closure of $N$ in $Δ$ is also equal to $N$; in addition, it is proved that any abelian normal subgroup of a finite totally $2$-closed group is cyclic; finally, it is proved that if a finite nilpotent group is a direct of two nilpotent subgroups where the two factors have coprime orders and both of them are totally 2-closed then G is …
We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables and let $G$ be a finite group for which the word map $w_G:G^d\rightarrow G$ has a fiber of size at least $ρ|G|^d$ for some fixed $ρ>0$. We show that, for certain words $w$, this implies that $G$ has a normal solvable subgroup of index bounded above in terms of $w$ and $ρ$. We also show that, for a larger family of words $w$, this implies that the nonsolvable length of $G$ is bounded above in terms of $w$ and $ρ$, thu…
We prove a result on the asymptotic proportion of randomly chosen pairs of permutations in the symmetric group $S_n$ which "invariably" generate a nonsolvable subgroup, i.e., whose cycle structures cannot possibly both occur in the same solvable subgroup of $S_n$. As an application, we obtain that for a large degree "random" integer polynomial $f$, reduction modulo two different primes can be expected to suffice to prove the nonsolvability of $Gal(f/\mathbb{Q})$.
There is a natural map from a symmetric group $S_n$ to a smaller symmetric group $S_{n-1}$, we write a decomposition of a permutation into a product of disjoint cycles and remove the element $n$ from this expression. For this reason there exists the inverse limit $\mathfrak{S}$ of sets $S_n$. We equip $S_n$ with the uniform distribution (or more generally with an Ewens distribution) and get a structure of a measure space on $\mathfrak{S}$ (it is called 'virtual permutations' or 'Chinese restaurant process'), a double $S_\infty\times S_\infty $ of an infinite symmetric group acts on $\mathfrak{…
This work provides a method(an algorithm) for solving the solvable unary algebraic equation $f(x)=0$ ($f(x)\in\mathbb{Q}[x]$) of arbitrary degree and obtaining the exact radical roots. This method requires that we know the Galois group as the permutation group of the roots of $f(x)$ and the approximate roots with sufficient precision beforehand. Of course, the approximate roots are not necessary but can help reduce the quantity of computation. The algorithm complexity is approximately proportional to the 4th power of the size of the Galois group of $f(x)$. The whole algorithm doesn't need to d…
Fix a positive integer $d$ and let $Γ_d$ be the class of finite groups without sections isomorphic to the alternating group $A_d$. The groups in $Γ_d$ were studied by Babai, Cameron and Pálfy in the 1980s and they determined bounds on the order of a primitive permutation group with this property, which have found a wide range of applications. Subsequently, results on the base sizes of such groups were also obtained. In this paper we replace the structural conditions on the group by restrictions on its point stabilizers, and we obtain similar, and sometimes stronger conclusions. For example, we…
To every involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation on a finite set $X$ there is a naturally associated finite solvable permutation group ${\mathcal G}(X,r)$ acting on $X$. We prove that every primitive permutation group of this type is of prime order $p$. Moreover, $(X,r)$ is then a so called permutation solution determined by a cycle of length $p$. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic soluti…
We refer to $d(G)$ as the minimal cardinality of a generating set of a finite group $G$, and say that $G$ is $d$-generated if $d(G)\leq d$. A transitive permutation group $G$ is called $\frac{3}{2}$-transitive if a point stabilizer $G_α$ is nontrivial and its orbits distinct from $\{α\}$ are of the same size. We prove that $d(G)\leq4$ for every primitive $\frac{3}{2}$-transitive permutation group $G$, moreover, $G$ is $2$-generated except for the very particular solvable affine groups that we completely describe. In particular, all finite $2$-transitive and $2$-homogeneous groups are $2$-gener…
A group $G$ is said to be totally $2$-closed if in each of its faithful permutation representations, say on a set $Ω$, $G$ is the largest subgroup of $\mathrm{Sym}(Ω)$ which leaves invariant each of the $G$-orbits for the induced action on $Ω\times Ω$. We prove that there are precisely $47$ finite totally $2$-closed groups with trivial Fitting subgroup. Each of these groups is a direct product of pairwise non-isomorphic sporadic simple groups, with the direct factors coming from the Janko groups $\mathrm{J}_1, \mathrm{J}_3$ and $\mathrm{J}_4$, together with $\mathrm{Ly}, \mathrm{Th}$ and the M…
