It is well known there is no finitely generated abelian group which has the R∞ property. We will show that also many non-finitely generated abelian groups do not have the R∞ property, but this does not hold for all of them! In fact we construct an uncountable number of infinite countable abelian groups which do have the R∞ property. We also construct an abelian group such that the cardinality o...

In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-abelian group G for which every non-linear irreducible complex representation is of degree 2 and with commutator subgroup G′ a central elementary abelian 2-group.

a group $g$ is called metahamiltonian if all its non-abelian subgroups are normal‎. ‎the aim of this paper is to provide an updated survey of research concerning certain classes of generalized metahamiltonian groups‎, ‎in various contexts‎, ‎and to prove some new results‎. ‎some open problems are listed‎.

Twist knots form a family of special two–bridge knots which include the trefoil knot and the figure eight knot. The knot group of a two–bridge knot has a particularly nice presentation with only two generators and a single relation. One could find our interest in this family of knots in the following facts: first, twist knots except the trefoil knot are hyperbolic; and second, twist knots are n...

The non-abelian tensor product of groups which has its origins in algebraic K-theory as well as inhomotopy theory, was introduced by Brown and Loday in 1987. Group theoretical aspects of non-abelian tensor products have been studied extensively. In particular, some studies focused on the relationship between the exponent of a group and exponent of its tensor square. On the other hand, com...

The problem of finding R-sequencings for abelian groups of even orders has been reduced to that of finding R∗sequencings for abelian groups of odd orders except in the case when the Sylow 2-subgroup is a non-cyclic non-elementary-abelian group of order 8. We partially address this exception, including all instances when the group has order 8t for t congruent to 1, 2, 3 or 4 (mod 7). As much is ...

In this article, by using a partial on locally compact semi-direct product groups, we present a compatible extension of the Fourier transform. As a consequence, we extend the fundamental theorems of Abelian Fourier transform to non-Abelian case.

Let FG be a group algebra of a finite non-abelian pgroup G and F a field of characteristic p. In this paper we give all minimal non-abelian p-groups and minimal non-metacyclic p-groups whose group algebras FG possess a filtered multiplicative F -basis.

Given any nontrivial alternating tri-character f on a finite abelian group G, one can construct a finite dimensional non-commutative and non-cocommutative semisimple Hopf algebra H. The group of group-like elements of H is an abelian central extension of B by Ĝ where B is the radical of f .