Narrow Results

Let $FG$ be the group algebra of a residually torsion free nilpotent group $G$ over a field $F$ of characteristic $0$. If $(x,y)$ is any pair of noncommuting elements of $G$, we show that for any rational number $r$ with $r\ne 0,\pm 1$, the subgroup $〈1+rx,1+ry,1+rxy〉$ is free of rank $3$ in the Malcev–Neumann field of fractions of $FG$. This result comes from a method of producing free groups of rank three in rational quaternions. In general, if a division ring $D$ has dimension $d^2$ over its center $Z$, and if the transcendence degree of $Z$ over its prime field ${𝔽}_p$ is $\ge d^2+3$, we present a method to construct free subgroups of rank three.

In this article, we introduce the normalizer of a subset X of a ring R (with identity) in the unit group and consider, in particular, the normalizer of the natural basis ±S of the integral semigroup ring ℤ₀S of a finite semigroup S. We investigate properties of this normalizer for the class of semigroup rings of inverse semigroups, which contains, for example, matrix rings, in particular, matrix rings over group rings, and partial group rings. We also construct free groups in the unit group of an integral semigroup ring of a Brandt semigroup using a bicyclic unit.

In this survey we revise the methods and results on the existence and construction of free groups of units in group rings, with special emphasis in integral group rings over finite groups and group algebras. We also survey results on constructions of free groups generated by elements which are either symmetric or unitary with respect to some involution and other results on which integral group rings have large subgroups which can be constructed with free subgroups and natural group operations.

Let $D$ be the field of fractions of the group algebra $k\mathrm{\Gamma }$ of the Heisenberg group $\mathrm{\Gamma }$, over the field $k$ of characteristic $\ne 2$. We show that for some involutions of $D$ that are not induced from involutions of $\mathrm{\Gamma }$, $D$ contains free symmetric and unitary pairs. We also give a general condition for a normal unitary subgroup of a division ring to contain a free group, and prove a generalization of Lewin’s Conjecture.

Let $D$ be a division ring with center $k$, $\mathrm{\text{char}}k=p\ne 2$, let ${}^{*}$ be an involution of $D$, and let $D^{†}$ be the multiplicative group of $D$. A pair $(u,v)$ is called free symmetric, if it is formed by symmetric elements, and it generates a free non-cyclic subgroup of $D^{†}$. If $U(L)$ is the enveloping algebra of the non-abelian nilpotent Lie $k$-algebra $L$ over the field $k$ of characteristic $\ne 2$, and ${}^{*}$ is a $k$-involution of $L$ extended to the field of fractions $D$ of $U(L)$, we show that $D^{†}$ contains free symmetric pairs. We also discuss the consequences of symmetric elements of a normal subgroup being torsion over the center.