Let $π$ be a set of primes. According to H. Wielandt, a subgroup $H$ of a finite group $X$ is called a $π$-submaximal subgroup if there is a monomorphism $ϕ:X\rightarrow Y$ into a finite group $Y$ such that $X^ϕ$ is subnormal in $Y$ and $H^ϕ=K\cap X^ϕ$ for a $π$-maximal subgroup $K$ of $Y$. In his talk at the well-known conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the $π$-submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set $π$ of primes, we obtain a description of the $π$-submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt's problem.
We explore the extent to which constructing the inductive theory of $\mathfrak{X}$-submaximal subgroups is possible. To this end, we study the behavior of $π$-submaximal subgroups under homomorphisms with $π$-separable kernels and construct examples where such behavior is irregular.
A nonempty class $\mathfrak{X}$ of finite groups is called complete if it is closed under taking subgroups, homomorphic images and extensions. We deal with a classical problem of determining $\mathfrak{X}$-maximal subgroups. We consider two definitions of submaximal $\mathfrak{X}$-subgroups suggested by Wielandt and discuss which one better suits our task. We prove that these definitions are not equivalent yet Wielandt-Hartley's theorem holds true for either definition of $\mathfrak{X}$-submaximality. We also give some applications of the strong version of Wielandt-Hartley's theorem.
Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the nonsolvable length of the group and denoted by $λ(G)$. For every integer $n$, we define a particular class of groups of nonsolvable length $n$, called \emph{$n$-rarefied}, and we show that every finite group of nonsolvable length $n$ contains an $n$-rarefied subgroup. As applications of this result, we improve the known upper bounds on $λ(G)$ and determine the max…
Let N be a minimal normal nonabelian subgroup of a finite group G. We will show that there exists a nontrivial irreducible character of N of degree at least 5 which is extendible to G. This result will be used to settle two open questions raised by Berkovich and Mann, and Berkovich and Zhmud'.
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 $Gal(f/\mathbb{Q})$.
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…
Let $G,D_{0},D_{1}$ be finite groups such that $D_{0}\trianglelefteq D_{1}$ are groups of automorphisms of $G$ that contain the inner automorphisms of $G$. Assume that $D_{1}/D_{0}$ has a normal $2$-complement and that $D_{1}$ acts fixed-point-freely on the set of $D_{0}$-conjugacy classes of involutions of $G$ (i.e., $C_{D_{1}}(a)D_{0}<D_{1}$ for every involution $a\in G$). We prove that $G$ is solvable. We also construct a nonsolvable finite group that possesses no characteristic conjugacy class of nontrivial cyclic subgroups. This shows that an assumption on the structure of $D_{1}/D_{0}$ a…
For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. This can be seen as an intermediate invariant between the probability that two elements generate a nilpotent subgroup and the probability that two elements generate a solvable subgroup. We prove that $sp(G) \leq 1/6$ for every nonsolvable group $G$.
Denote by $ω(G)$ the number of orbits of the action of $Aut(G)$ on the finite group $G$. We prove that if $G$ is a finite nonsolvable group in which $ω(G) \leqslant 5$, then $G$ is isomorphic to one of the groups $A_5,A_6,PSL(2,7)$ or $PSL(2,8)$. We also consider the case when $ω(G) = 6$ and show that if $G$ is a nonsolvable finite group with $ω(G) = 6$, then either $G \simeq PSL(3,4)$ or there exists a characteristic elementary abelian $2$-subgroup $N$ of $G$ such that $G/N \simeq A_5$.
Let $G$ be a group. The orbits of the natural action of $\mbox{Aut}(G)$ on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $ω(G)$. In this paper the finite nonsolvable groups $G$ with $ω(G) \leq 6$ are classified - this solves a problem posed by Markus Stroppel - and it is proved that there are infinitely many finite nonsolvable groups $G$ with $ω(G)=7$. Moreover it is proved that for a given number $n$ there are only finitely many finite groups $G$ without nontrivial abelian normal subgroups and such that $ω(G) \leq n$, generalizing a re…
This work is a continuation of Automorphisms of $K$-groups I, P. Flavell, preprint. The main object of study is a finite $K$-group $G$ that admits an elementary abelian group $A$ acting coprimely. For certain group theoretic properties $\mathcal P$, we study the $AC_{G}(A)$-invariant $\mathcal P$-subgroups of $G$. A number of results of McBride, 'Near solvable signalizer functors on finite groups' J. Algebra {\bf 78}(1) (1982) 181-214 and 'Nonsolvable signalizer functors on finite groups', J. Algebra {\bf 78}(1) (1982) 215-238 are extended. One purpose of this work is to build a general theory…
Let $A$ be an elementary abelian $r$-group with rank at least $3$ that acts faithfully on the finite $r'$-group $G$. Assume that $G$ is $A$-simple, so that $G = K_{1} \times\cdots\times K_{n}$ where $K_{1},\ldots,K_{n}$ is a collection of simple subgroups of $G$ that is permuted transitively by $A$. The purpose of this paper is to characterize $G$ and the collection of fixed point subgroups $\{ C_{G}(a) \;|\; a \in A^{\#} \}$. An application of this result will be a new proof of McBride's Nonsolvable Signalizer Functor Theorem.
We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either $G$ is a $p$-group or $G = PN$ where $P$ is a Sylow $p$-subgroup and $(G,P,P \cap N)$ is a Frobenius-Wielandt triple. We also prove that if all the elements of $G \setminus N$ have prime power orders and the orders are divisible by two primes $p$ and $q$, then $G$ is a $\{ p, q \}$-group and $G/N$ is either a Frobenius group or a $2$-Frobenius group. If all th…
We show that a nonempty family of $n$-generated subgroups of a pro-$p$ group has a maximal element. This suggests that 'Noetherian Induction' can be used to discover new features of finitely generated subgroups of pro-$p$ groups. To demonstrate this, we show that in various pro-$p$ groups $Γ$ (e.g. free pro-$p$ groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup $H \neq 1$ is the greatest subgroup of $Γ$ containing $H$ as an open subgroup. We also show that an ascending sequence of $n$-generated subgroups of a limit group must terminate (this extends the an…
In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition: $G$ is solvable if for all conjugacy classes $C$ and $D$ of $G$, \emph{there exist} $x\in C$ and $y\in D$ for which $\gen{x,y}$ is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if $G$ is a finite nonabelian simple group, then there exist two integers $a$ an…
We show that if G is a finite group then no chain of modular elements in its subgroup lattice L(G) is longer than a chief series. Also, we show that if G is a nonsolvable finite group then every maximal chain in L(G) has length at least two more than that of the chief length of G, thereby providing a converse of a result of J. Kohler. Our results enable us to give a new characterization of finite solvable groups involving only the combinatorics of subgroup lattices. Namely, a finite group G is solvable if and only if L(G) contains a maximal chain X and a chain M consisting entirely of modular …
A loop $(X,\circ)$ is said to be a Bruck loop if it satisfies the (right) Bol identity $((z\circ x)\circ y)\circ x = z\circ ((x\circ y)\circ x)$ and the automorphic inverse property $(x\circ y)^{-1}=x^{-1}\circ y^{-1}$. If $X$ is a finite Bruck loop and $G$ is the group generated by all right translations $R(x): y\mapsto y\circ x$, then we show that $X$ and $G$ are central products $X = O^{2'}(X) * O(X)$ and $G = O^{2'}(G) * O(G)$, where $O^{2'}(X)$ ($O^{2'}(G)$) is the subloop (subgroup) generated by all 2-elements, and $O(X)$ ($O(G)$) is the largest normal subloop (subgroup) of odd order. In…
