1. Introduction. Let p be a prime number such that q — l{p — l) is also a prime. Let Q, be the set of symbols 1, • • • , p, and ® be a non-solvable transitive permutation group on 12. Such permutation groups were first considered by Galois in 1832 [I, §327; III, §262]: if the linear fractional group LF 2 (l) over the field of I elements, where I is a prime number not smaller than five, contains...

Let G be a reductive algebraic group defined over an algebraically closed field k. Let H be a closed connected subgroup of G containing a maximal torus T of G. In [13] it was shown (at least in characteristic zero) that the parabolic subgroups of G can be characterized among all such subgroups H by a certain finiteness property of the induction functor (-)Iz and its derived functors Lk,G(-). Th...

We study two group theoretic problems, Group Intersection and Double Coset Membership, in the setting of black-box groups, where Double Coset Membership generalizes a set of problems, including Group Membership, Group Factorization, and Coset Intersection. No polynomial-time classical algorithms are known for these problems. We show that for solvable groups, there exist efficient quantum algori...

Let $ G$ be a finite group and $p$ prime. \mathrm{Vo}(G) denote the set of orders vanishing elements, $\mathrm{Vo}_{p} (G)$ subset consisting those elements divisible by $\mathrm{Vo}_{p'} (G) not $p$. Dolfi, Pacifi, Sanus Spiga proved that if is p $-power for all a\in \mathrm{Vo}(G)$, then G has normal Sylow $-subgroup. In another article, same authors also show \mathrm{Vo}_{p'}(G) =\emptyset $...

In this paper we consider Q the class of solvable Lie algebras L with the following property: if A is a subalgebra of L, then Φ(A) ⊆ Φ(L) (where Φ(L) denotes the Frattini subalgebra of L;that is Φ(L) is the intersection of all maximal subalgebras of L). The class Q is shown to contain all solvable Lie algebras whose derived algebra is nilpotent. Necessary conditions are found such that an ideal...

Following Malcev [5], we will call a subgroup M of a group G finitely separable if for any element g ∈ G, not belonging to M, there exists a homomorphism φ of group G onto some finite group X such that gφ / ∈Mφ. This is equivalent to the statement that for any element g ∈ G \M, there exists a normal subgroup N of finite index in G such that g / ∈MN . A group G is subgroup separable if each of i...

With every nontrivial connected algebraic group G we associate a positive integer gtd(G) called the generic transitivity degree of G and equal to the maximal n such that there is a nontrivial action of G on an irreducible algebraic variety X for which the diagonal action of G on Xn admits an open orbit. We show that gtd(G) 6 2 (respectively, gtd(G) = 1) for all solvable (respectively, nilpotent...

Abstract We prove that there exists a universal constant D such if p is prime divisor of the index Fitting subgroup finite group G , then number conjugacy classes at least $Dp/\log_2p$ . conjecture we can take $D=1$ and for solvable groups, $D=1/3$

In a 2001 paper, Shareshian conjectured that the subgroup lattice of a finite, solvable group has an EL-labeling. We construct such a labeling and verify that our labeling has the expected properties.