In this paper we describe some algorithms to identify permutable and Sylow-permutable subgroups of finite groups, Dedekind and Iwasawa finite groups, and finite T-groups (groups in which normality is transitive), PT-groups (groups in which permutability is transitive), and PST-groups (groups in which Sylow permutability is transitive). These algorithms have been implemented in a package for the computer algebra system GAP.
M. Bestvina has shown that for any given torsion-free CAT(0) group G, all of its boundaries are shape equivalent. He then posed the question of whether they satisfy the stronger condition of being cell-like equivalent. In this article we prove that the answer is "Yes" in the situation where the group in question splits as a direct product with infinite factors. We accomplish this by proving an interesting theorem in shape theory.
In Universal Algebra, identities are used to classify algebras into collections, called varieties and hyperidentities are use to classify varieties into collections, called hypervarities. The concept of a hypersubstitution is a tool to study hyperidentities and hypervarieties. Generalized hypersubstitutions and strong identities generalize the concepts of a hypersubstitution and of a hyperidentity, respectively. The set of all generalized hypersubstitutions forms a monoid. In...
We prove that with probability tending to 1, a one-relator group with at least three generators and the relator of length is residually finite, is a virtually residually (finite -)group for all sufficiently large , and is coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions.
Characterizations of 'almost associative' binary operations generating a minimal clone are given for two interpretations of the term 'almost associative'. One of them uses the associative spectrum, the other one uses the index of nonassociativity to measure how far an operation is from being associative.
Generalizing the notion of the almost free group we introduce almost Butler groups. An almost -group of singular cardinality is a -group. Since almost -groups have preseparative chains, the same result in regular cardinality holds under the additional hypothesis that is a -group. Some other results characterizing -groups within the classes of almost -groups and almost -groups are obtained. A theorem of stating that a group of weakly compact cardinality having a -filtration consisting...
A new class of -primary abelian groups that are Hausdorff in the -adic topology and that generalize direct sums of cyclic groups are studied. We call this new class of groups almost coproducts of cyclic groups. These groups are defined in terms of a modified axiom 3 system, and it is observed that such groups appear naturally. For example, is almost a coproduct of finite cyclic groups whenever is a Hausdorff -primary group and is the group of normalized units of the modular group algebra...
Let ϕ be an automorphism of prime order p of the group G with C G(ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are...
Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that . For simplicity we will call such modules splitters (see [10]). Also other names like stones are used (see a dictionary in Ringel’s paper [8]). Our investigation continues [5]. In [5] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [5] we concentrated on splitters which are larger...