We show that for any finite group G and for any d there exists a word w ∈ Fd such that a d-tuple in G satisfies w if and only if it generates a solvable subgroup. In particular, if G itself is not solvable, then it cannot be obtained as a quotient of the one relator group Fd/ 〈w〉. As a corollary, the probability that a word is satisfied in a fixed nonsolvable group can be made arbitrarily small...

We examine the structure of the Levi component MA in a minimal parabolic subgroup P = MAN of a real reductive Lie group G and work out the cases where M is metabelian, equivalently where p is solvable. When G is a linear group we verify that p is solvable if and only if M is commutative. In the general case M is abelian modulo the center ZG , we indicate the exact structure of M and P , and we ...

We prove that if Γ is subgroup of Diff 1+ + (I) and N is a natural number such that every non-identity element of Γ has at most N fixed points then Γ is solvable. If in addition Γ is a subgroup of Diff +(I) then we can claim that Γ is metaabelian. It is a classical result (essentially due to Hölder, cf.[N]) that if Γ is a subgroup of Homeo+(R) such that every nontrivial element acts freely then...

In §1 we use COO-vector methods, essentially Frobenius reciprocity, to derive the Howe-Richardson multiplicity formula for compact nilmanifolds. In §2 we use Frobenius reciprocity to generalize and considerably simplify a reduction procedure developed by Howe for solvable groups to general extensions of nilpotent Lie groups. In §3 we give an application of the previous results to obtain a reduc...

Let k0 be a finite algebraic number field in a fixed algebraic closure Ω and ζn denote a primitive n-th root of unity ( n ≥ 1). Let k∞ be the maximal cyclotomic extension of k0, i.e. the field obtained by adjoining to k0 all ζn ( n = 1, 2, ...). Let M and L be the maximal abelian extension of k∞ and the maximal unramified abelian extension of k∞ respectively. The Galois groups Gal(M/k∞) and Gal...

For an irreducible complex character χ of the finite group G, let π(χ) denote set prime divisors degree χ(1) χ. Denote then by ρ(G) union all sets and σ(G) largest value |π(χ)|, as runs in Irr(G). The ρ-σ conjecture, formulated Bertram Huppert 80's, predicts that |ρ(G)|≤3σ(G) always holds, whereas |ρ(G)|≤2σ(G) holds if G is solvable; moreover, O. Manz T.R. Wolf proposed a “strengthened” form co...

If G is a finite p-solvable group for some prime p, A a solvable subgroup of the automorphism group of G of order prime to |G| such that A stabilises a p-block b of G and acts trivially on a defect group P of b, then there is a Morita equivalence between the block b and its Watanabe correspondent w(b) of CG(A), given by a bimodule M with vertex ∆P and an endo-permutation module as source, which...

solvable graphs (also known as reachable graphs) are types of graphs that any arrangement of a specified number of agents located on the graph’s vertices can be reached from any initial arrangement through agents’ moves along the graph’s edges, while avoiding deadlocks (interceptions). in this paper, the properties of solvable graphs are investigated, and a new concept in multi agent moti...

Malle proposed a conjecture for counting the number of $G$-extensions $L/K$ with discriminant bounded above by $X$, denoted $N(K,G;X)$, where $G$ is fixed transitive subgroup $G\subset S_n$ and $X$ tends towards infinity. We introduce refinement Malle's conjecture, if group nontrivial Galois action then we consider set crossed homomorphisms in $Z^1(K,G)$ (or equivalently $1$-coclasses $H^1(K,G)...

A Brauer character of a finite group may be lifted to an ordinary character if it lies in a block whose defect groups are contained in a normal p-solvable subgroup. By the Fong-Swan theorem [2, Theorem 72.1], an irreducible Brauer character of a finite p-solvable group G may be lifted to an ordinary (complex) character of G. In other words, every Brauer character is the restriction of some ...