For any prime number $p$ and field $k$, we characterize the $p$-retract rationality of an algebraic $k$-torus in terms of its character lattice. We show that a $k$-torus is retract rational if and only if it is $p$-retract rational for every prime $p$, and that the Noether problem for retract rationality for a group of multiplicative type $G$ has an affirmative answer for $G$ if and only if the Noether problem for $p$-retract rationality for $G$ has a positive answer for all $p$. For every finite set of primes $S$ we give examples of tori that are $p$-retract rational if and only if $p\notin S$.

We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus conjecture for the group PSL(2, 19). We also prove the Zassenhaus conjecture for PSL(2, 23). In a second application we show that there are no normalized units of order 6 in the integral group rings of M 10 and PGL(2, 9). This completes the proof of a theorem of Kimmerle and Konovalov that shows that the prime graph question has an affirmative answer for all groups having an order divisible by at most three different primes.

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$ -group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$ -adic vector space associated with  $M$ . This leads to our second variation, which is a study of the finite linear groups that can arise.

From the mid-1990s onwards, the main focus of L. G. Kovács’ research was on Lie powers. This brief survey presents some of the key results on Lie powers obtained by Kovács and his collaborators, and discusses some subsequent developments and applications of this work.

We consider the problem of whether the norm one torus defined by a finite separable field extension K/k is stably (or retract) rational over k. This has already been solved for the case where K/k is a Galois extension. In this paper, we solve the problem for the case where K/k is a non-Galois extension such that the Galois group of the Galois closure of K/k is nilpotent or metacyclic.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let 𝔤 be its Lie algebra. Let k(G), respectively, k(𝔤), be the field of k-rational functions on G, respectively, 𝔤. The conjugation action of G on itself induces the adjoint action of G on 𝔤. We investigate the question whether or not the field extensions k(G)/k(G)G and k(𝔤)/k(𝔤)G are purely transcendental. We show that the answer is the same for k(G)/k(G)G and k(𝔤)/k(𝔤)G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type An or Cn, and negative for groups of other types, except possibly G2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let ${{G}_{0}}$ and ${{G}_{1}}$ be countable abelian groups. Let ${{\gamma }_{i}}$ be an automorphism of ${{G}_{i}}$ of order two. Then there exists a unital Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and with $[{{1}_{A}}]\,=\,0$ in ${{K}_{0}}(A)$, and an automorphism $\alpha \,\in \,\text{Aut}(A)$ of order two, such that ${{K}_{0}}(A)\,\cong \,{{G}_{0}}$, such that ${{K}_{1}}(A)\,\cong \,{{G}_{1}}$, and such that ${{\alpha }_{*}}\,:\,{{K}_{i}}(A)\,\to \,{{K}_{i}}(A)$ is ${{\gamma }_{i}}$. As a consequence, we prove that every ${{\mathbb{Z}}_{2}}$-graded countable module over the representation ring $R({{\mathbb{Z}}_{2}})$ of ${{\mathbb{Z}}_{2}}$ is isomorphic to the equivariant $K$-theory ${{K}^{{{\mathbb{Z}}_{2}}}}(A)$ for some action of ${{\mathbb{Z}}_{2}}$ on a unital Kirchberg algebra $A$.

Along the way, we prove that every not necessarily finitely generated $\mathbb{Z}\left[ {{\mathbb{Z}}_{2}} \right]$-module which is free as a $\mathbb{Z}$-module has a direct sum decomposition with only three kinds of summands, namely $\mathbb{Z}\left[ {{\mathbb{Z}}_{2}} \right]$ itself and $\mathbb{Z}$ on which the nontrivial element of ${{\mathbb{Z}}_{2}}$ acts either trivially or by multiplication by −1.

Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. $A$ is defined over a subfield $F_0$ if there exists an $F_0$-algebra $A_0$ such that $A = A_0 \bigotimes_{F_0} F$. The following are shown. (i) In many cases $A$ can be defined over a rational extension of $k$. (ii) If $A$ has odd degree $n \geq 5$, then $A$ is defined over a field $F_0$ of transcendence degree $\leq {\frac{1}{2}}(n-1)(n-2)$ over $k$. (iii) If $A$ is a $\mathbb{Z}/m \times \mathbb{Z}/2$-crossed product for some $m \geq 2$ (and in particular, if $A$ is any algebra of degree 4) then $A$ is Brauer equivalent to a tensor product of two symbol algebras. Consequently, ${\rm M}_m(A)$ can be defined over a field $F_0$ such that ${\rm trdeg}_k(F_0) \leq 4$. (iv) If $A$ has degree 4 then the trace form of $A$ can be defined over a field $F_0$ of transcendence degree $\leq 4$. (In (i), (iii) and (iv) it is assumed that the center of $A$ contains certain roots of unity.)

If $G$ is a finite group and $V$ is a finite-dimensional ${\rm Q\!\!\!I}\,[G]$-module, $V$ is isomorphic to its contragredient module $V^*$. In general, $V$ need not contain any ${\bb Z}[G]$-lattice which is locally isomorphic to its contragredient lattice. Nevertheless, it turns out that for every $V$ there exists another ${\rm Q\!\!\!I}\,[G]$-module $V^{\prime}$; such that both $V^{\prime}$; and $V \oplus V^{\prime}$; contain ${\bb Z}[G]$-lattices which are locally isomorphic to their contragredient lattices.