We characterise finite unitary rings $R$ such that all Sylow subgroups of the group of units $R^{\ast }$ are cyclic. To be precise, we show that, up to isomorphism, $R$ is one of the three types of rings in $\{O,E,O\oplus E\}$, where $O\in \{GF(q),\mathbb{Z}_{p^{\unicode[STIX]{x1D6FC}}}\}$ is a ring of odd cardinality and $E$ is a ring of cardinality $2^{n}$ which is one of seven explicitly described types.

A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.

In this paper we find all finite rings with a nilpotent group of units. It was thought that the answer to this was already given by McDonald in 1974, but as was shown by Groza in 1989, the conclusions that had been reached there do not hold. Here, we improve some results of Groza and describe the structure of an arbitrary finite ring with a nilpotent group of units, thus solving McDonald’s problem.

Let F be a field of characteristic p and G a group containing at least one element of order p. It is proved that the group of units of the group algebra FG is a bounded Engel group if and only if FG is a bounded Engel algebra, and that this is the case if and only if G is nilpotent and has a normal subgroup H such that both the factor group G/H and the commutator subgroup H′ are finite p–groups.

Let $D$ be a division ring whose group of units satisfies a non-trivial group identity $w$. Let $\alpha$ be the sum of positive degrees of indeterminates occurring in $w$. If the centre of $D$ contains more than $3\alpha$ elements, then $D$ is commutative.

AMS 2000 Mathematics subject classification: Primary 16R50. Secondary 16K20

Let p be a prime, a field of pn elements, and G a finite p-group. It is shown here that if G has a quotient whose commutator subgroup is of order p and whose centre has index pk, then the group of normalized units in the group algebra has a conjugacy class of pnk elements. This was first proved by A. Bovdi and C. Polcino Milies for the case k = 2; their argument is now generalized and simplified. It remains an intriguing question whether the cardinality of the smallest noncentral conjugacy class can always be recognized from this test.

Let G be a group, let RG be the group ring of the group G over the unital commutative ring R and let U(RG) be its group of units. Conditions which imply that U(RG) contains no free noncyclic group are studied, when R is a field of characteristic p ≠ 0, not algebraic over its prime field, and G is a solvable-by-finite group without p-elements. We also consider the case R = ℤp, the ring of p-adic integers and G torsionby- nilpotent torsion free group. Finally, the residual nilpotence of U(ℤpG) is investigated.

In this paper we give necessary and sufficient conditions under which the group of units of a group ring of a finite group G over a field K does not contain a free subgroup of rank 2.

Some extensions of this results to infinite nilpotent and FC groups are also considered.