The present paper is a note on the relative tensor degree of finite groups. This notion generalizes the tensor degree, introduced recently in literature, and allows us to adapt the concept of relative commutativity degree through the notion of nonabelian tensor square. We show two inequalities, which correlate the relative tensor degree with the relative commutativity degree of finite groups. 1...

For an associative ring R, let P be an R-module with S = EndR(P). C. Menini and A. Orsatti posed the question of when the related functor HomR(P, −) (with left adjoint P ⊗S −) induces an equivalence between a subcategory of R M closed under factor modules and a subcategory of S M closed under submodules. They observed that this is precisely the case if the unit of the adjunction is an epimorphi...

For many decision problems on a finitely presented group G, we can quickly weed out negative solutions by using much quicker algorithms on an appropriately chosen quotient group G/K of G. However, the behavior of such “quotient tests” can be sometimes paradoxical. In this paper, we analyze a few simple case studies of quotient tests for the classical identity, word, conjugacy problems in groups...

The problem of proving that a given finitely presented group G is infinite can be very difficult (or impossible) in general, but there are several possible approaches that can be tried, some of which can be rendered much more effective with the assistance of computers. To this list, we add another one, which can be applied if an epimorphism of G onto a finite group H is known. The idea depends ...

A group epimorphism Q: G+ Q preserves perfect radicals if $PG = PQ, where PC, PQ is maximal perfect in G, Q respectively. The following two questions are considered. Question 1. If P,,(B) acts trivially on H,(F; E), does the tibration F + Es B then have ~,(P)(P~,(E)) = Pm,(B)? Question 2. If f: X + Y is a map of spaces which induces an isomorphism of homology groups, is rr,(X) * rr,( Y) n,( Y)/...

Let A be an Ä-order in a semisimple finite dimensional /(-algebra, where K is an algebraic number field, and R is the ring of algebraic integers of K. Denote by C(A) the reduced class group of the category of locally free left A-lattices. Choose A= ZC, the integral group ring of a finite group G, and let A be a maximal Z-order in QG containing A. There is an epimorphism C(A)-C(A'), given by JIÍ...

Let K be the kernel of an epimorphism χ : G → Z, for G a finitely presented group. If K has uncountably many normal subgroups of finite index r, then K has uncountably many subgroups (not necessarily normal) of any finite index greater than r. In particular, this is the case whenever G is subgroup separable and K is nonfinitely generated. Assume that G has an abelian HNN base contained in K. If...

by a quasi-permutation matrix we mean a square matrix over the complex field c with non-negative integral trace. thus, every permutation matrix over c is a quasipermutation matrix. for a given finite group g, let p(g) denote the minimal degree of a faithful permutation representation of g (or of a faithful representation of g by permutation matrices), let q(g) denote the minimal degree of a fai...