We study three classes of subgroups of Polish groups: Borel subgroups, Polishable subgroups, and maximal divisible subgroups. The membership of a subgroup in each of these classes allows one to assign to it a rank, that is, a countable ordinal, measuring in a natural way complexity of the subgroup. We prove theorems comparing these three ranks and construct subgroups with prescribed ranks. In p...

The aim of the present paper is to prove the following result: Let (#t)t>0 be a vaguely cont inuous convolut ion semigroup consisting of symmetric probabil i ty measures on a locally compact abelian group G. Then there exists a closed subgroup H of G such that supp (#t) = H for all t > 0. As a consequence of this result we get that the support of a symmetric infinitely divisible distribution in...

Let p be a prime and G a subgroup of GLd(p). We define G to be pexceptional if it has order divisible by p, but all its orbits on vectors have size coprime to p. We obtain a classification of p-exceptional linear groups. This has consequences for a well known conjecture in representation theory, and also for a longstanding question concerning 1 2 -transitive linear groups (i.e. those having all...

The covering number of a finite group [Formula: see text], denoted is the smallest positive integer text] such that union proper subgroups. We calculate for family primitive groups with unique minimal normal subgroup isomorphic to divisible by and cyclic. This generalization result Swartz concerning symmetric groups. also prove an asymptotic pairwise generation.

The aim of this paper is to present a construction of t-divisible designs for t > 3, because such divisible designs seem to be missing in the literature. To this end, tools such as finite projective spaces and their algebraic varieties are employed. More precisely, in a first step an abstract construction, called t-lifting, is developed. It starts from a set X containing a tdivisible design and...

Let L be a commutative Moufang loop (CML) with multiplication group M, and let F(L), F(M) be the Frattini subgroup and Frattini subgroup of L and M respectively. It is proved that F(L) = L if and only if F(M) = M and is described the structure of this CLM. Constructively it is defined the notion of normalizer for subloops in CML. Using this it is proved that if F(L) 6= L then L satisfies the no...

We show that the automorphism group of a divisible design D is isomorhic to a subgroup H of index 1 or 2 in the automorphism group Aut C(D) of the associated constant weight code. Only in very special cases, H is not the full automorphism group.

We describe definable relations in the real field augmented by a binary relation which is an arbitrary multiplicative group of complex numbers contained in the divisible hull of a finitely generated subgroup of the unit circle. We give a complete axiom system for this structure which admits quantifier elimination down to Boolean combinations od existential formulas of a special

We study finite groups G with the property that for any subgroup M maximal in whose order is divisible by all prime divisors of |G|, supersolvable. show nonabelian simple group can occur as a composition factor such and that, if solvable, then nilpotency length rank are arbitrarily large. On other hand, every p, p-length at most 1. This answers questions proposed V. Monakhov The Kourovka Notebo...