Let $S$ be a finitely generated pro-$p$ group. Let ${\mathcal{E}}_{p^{\prime }}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects nontrivially with every nontrivial normal subgroup of $G$. In this paper, we investigate whether or not there is a bound on $|G:S|$ for $G\in {\mathcal{E}}_{p^{\prime }}(S)$. For instance, we give an example where ${\mathcal{E}}_{p^{\prime }}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $|G:S|$ is bounded in the case where $S$ is just infinite.

We show that two metacyclic groups of the following types are isomorphic if they have the same character tables: (i) split metacyclic groups, (ii) the metacyclic p-groups and (iii) the metacyclic {p, q}-groups, where p, q are odd primes.

The Wielandt subgroup of a group $G$, denoted by ${\it\omega}(G)$, is the intersection of the normalisers of all subnormal subgroups of $G$. The terms of the Wielandt series of $G$ are defined, inductively, by putting ${\it\omega}_{0}(G)=1$ and ${\it\omega}_{i+1}(G)/{\it\omega}_{i}(G)={\it\omega}(G/{\it\omega}_{i}(G))$. In this paper, we investigate the relations between the$p$-length of a $p$-soluble finite group and the Wielandt series of its Sylow $p$-subgroups. Some recent results are improved.

We introduce support varieties for rational representations of a linear algebraic group $G$ of exponential type over an algebraically closed field $k$ of characteristic $p>0$. These varieties are closed subspaces of the space $V(G)$ of all 1-parameter subgroups of $G$. The functor $M\mapsto V(G)_{M}$ satisfies many of the standard properties of support varieties satisfied by finite groups and other finite group schemes. Furthermore, there is a close relationship between $V(G)_{M}$ and the family of support varieties $V_{r}(G)_{M}$ obtained by restricting the $G$ action to Frobenius kernels $G_{(r)}\subset G$. These support varieties seem particularly appropriate for the investigation of infinite-dimensional rational $G$-modules.

Let ${\rm\Delta}:G\rightarrow \text{GL}(n,K)$ be an absolutely irreducible representation of an arbitrary group  $G$ over an arbitrary field  $K$ ; let ${\it\chi}:G\rightarrow K:g\mapsto \text{tr}({\rm\Delta}(g))$ be its character. In this paper, we assume knowledge of ${\it\chi}$ only, and study which properties of ${\rm\Delta}$ can be inferred. We prove criteria to decide whether ${\rm\Delta}$ preserves a form, is realizable over a subfield, or acts imprimitively on  $K^{n\times 1}$ . If $K$ is finite, we can decide whether the image of ${\rm\Delta}$ belongs to certain Aschbacher classes.

We prove a cohomological property for a class of finite $p$-groups introduced earlier by Xu, which we call semi-abelian $p$-groups. This result implies that a semi-abelian $p$-group has noninner automorphisms of order $p$, which settles a long-standing problem for this class. We answer also, independetly, an old question posed by Xu about the power structure of semi-abelian $p$-groups.

The classes of finite groups with minimal sets of generators of fixed cardinalities, named ${\mathcal{B}}$-groups, and groups with the basis property, in which every subgroup is a ${\mathcal{B}}$-group, contain only $p$-groups and some $\{p,q\}$-groups. Moreover, abelian ${\mathcal{B}}$-groups are exactly $p$-groups. If only generators of prime power orders are considered, then an analogue of property ${\mathcal{B}}$ is denoted by ${\mathcal{B}}_{pp}$ and an analogue of the basis property is called the pp-basis property. These classes are larger and contain all nilpotent groups and some cyclic $q$-extensions of $p$-groups. In this paper we characterise all finite groups with the pp-basis property as products of $p$-groups and precisely described $\{p,q\}$-groups.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.

A Schunck class $\mathfrak{H}$ is determined by the class $\mathfrak{X}$ of primitives contained in $\mathfrak{H}$. We give necessary and sufficient conditions on $\mathfrak{X}$ for $\mathfrak{H}$ to be a saturated formation.

Let G be a finite p-solvable group and P a Sylow p-subgroup of G. Suppose that $\gamma_{\ell (p-1)}(P)\subseteq \gamma_r(P)^{p^s}$ for ℓ(p−1) < r + s(p − 1), then the p-length is bounded by a function depending on ℓ.

We describe how to compute the ideal class group and the unit group of an order in a number field in subexponential time. Our method relies on the generalized Riemann hypothesis and other usual heuristics concerning the smoothness of ideals. It applies to arbitrary classes of number fields, including those for which the degree goes to infinity.

Assume that $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}m$ and $n$ are two positive integers which do not divide each other. If the set of conjugacy class sizes of primary and biprimary elements of a group $G$ is $\{1, m, n, mn\}$, we show that up to central factors $G$ is a $\{p,q\}$-group for two distinct primes $p$ and $q$.

We prove the existence of certain rationally rigid triples in ${E}_{8}(p)$ for good primes $p$ (i.e. $p>5$) thereby showing that these groups occur as Galois groups over the field of rational numbers. We show that these triples arise from rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic zero. As a byproduct of the proof, we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a short list of possible overgroups of regular unipotent elements in simple exceptional groups.

We show that the space of actions of every finitely generated, nilpotent group by $C^1$ orientation-preserving diffeomorphisms of the circle is path-connected. This is done via a general result that allows any given action on the interval to be connected to the trivial one by a continuous path of topological conjugates.

We study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ as in the tame case. Also, we link a crepant resolution with a count of Artin–Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

It is shown that if $G$ is a finite $p$-group of coclass 2 with $p>2$, then $G$ has a noninner automorphism of order $p$.

We study the action of the formal affine Hecke algebra on the formal group algebra, and show that the the formal affine Hecke algebra has a basis indexed by the Weyl group as a module over the formal group algebra. We also define a concept called the normal formal group law, which we use to simplify the relations of the generators of the formal affine Demazure algebra and the formal affine Hecke algebra.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a finite 2-group. If $G$ is of coclass 2 or $(G,Z(G))$ is a Camina pair, then $G$ admits a noninner automorphism of order 2 or 4 leaving the Frattini subgroup elementwise fixed.

A subgroup $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ of a finite group $G$ is said to be S-semipermutable in $G$ if $H$ permutes with every Sylow $q$-subgroup of $G$ for all primes $q$ not dividing $|H |$. A finite group $G$ is an MS-group if the maximal subgroups of all the Sylow subgroups of $G$ are S-semipermutable in $G$. The aim of the present paper is to characterise the finite MS-groups.