We determine the torsion subgroup of the group of endotrivial modules for a finite solvable group in characteristic p. We also prove that our result would hold for p-solvable groups, provided a conjecture can be proved about the case of p-nilpotent groups.

We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x ∈ G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite dimensional Lie algebras and some classes of infinite groups. We also consider a similar problem for pairs of elements.

We give an example of a residually-p finitely generated group, that satisfies a non-trivial group law, but is not virtually solvable. Denote by Fn the free group on n generators. Recall that, given a group word m(x1, . . . , xn) ∈ Fn, a group G satisfies the law m = 1 if for every u1, . . . , un ∈ G, m(u1, . . . , un) = 1. Given a set S of group laws, the n-generator free group in the variety g...

Let φ be Euler’s phi function. We prove that a vertex-transitive graph 0 of order n, with gcd(n, φ(n)) = 1, is isomorphic to a circulant graph of order n if and only if Aut(0) contains a transitive solvable subgroup. As a corollary, we prove that every vertex-transitive graph 0 of order n is isomorphic to a circulant graph of order n if and only if for every such 0, Aut(0) contains a transitive...

We give efficient quantum algorithms for HIDDEN TRANSLATION and HIDDEN SUBGROUP in a large class of non-abelian groups including solvable groups of bounded exponent and of bounded derived series. Our algorithms are recursive. For the base case, we solve efficiently HIDDEN TRANSLATION in Z p , whenever p is a fixed prime. For the induction step, we introduce the problem ORBIT COSET generalizing ...

We prove that there exists a constant k with the property: if C is a conjugacy class of a finite group G such that every k elements of C generate a solvable subgroup then C generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take k = 4. We also present proofs that do not use the Classification theorem. The most direct proof gives a...

Let Γ be a finitely generated residually finite group. Denote by sn(Γ) (resp. tn(Γ)) the number of subgroups (resp. normal subgroups) of Γ of index at most n. In the last two decades the study of the connection between the algebraic structure of Γ and the growth rate of the sequence {sn(Γ)}n=1 has become a very active area of research under the rubric “subgroup growth” (see [L1], [LS] and the r...

An abundance of information regarding the structure of a finite group can be obtained by studying its irreducible characters. Of particular interest are monomial characters — those induced from a linear character of some subgroup — since Brauer has shown that any irreducible character of a group can be written as an integral linear combination of monomial characters. Our primary focus is the cl...

Let S be finite nonempty set of inequivlent valuations on Fp(t), and OS be the ring of S-integers. If Bn is the solvable, linear algebraic group of upper triangular matrices with determinant 1, then the solvable S-arithmetic group Bn(OS) has a finite index subgroup with infinite dimensional cohomology group in dimension |S|.

In the discussion of A-solvable groups, the question arises if a torsion-free abelian group A of finite rank is flat as a module over its endomorphism ring if every A-generated subgroup of a torsion-free A-solvable group is A-solvable. This paper gives a negative answer by constructing a torsion-free group of rank 3 for which all A-generated torsion-free groups are A-solvable, although A is not...