Related papers: Finite group schemes of $p$-rank $\leq1$

We investigate the saturation rank of a finite group scheme, defined over an algebraically closed field $\Bk$ of positive characteristic $p$. We begin by exploring the saturation rank for finite groups and infinitesimal group schemes.…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in…

We give a definition of full level structure on group schemes of the form $G\times G$, where $G$ is a finite flat commutative group scheme of rank $p$ over a $\mathbb{Z}_p$-scheme $S$ or, more generally, a truncated $p$-divisible group of…

Let $G$ be a special $p$-group. If $G$ is of rank two, or $G$ is of maximum rank with $|G^p|\leq p$, then we describe the complex irreducible projective representations of $G$.

The `upper rank' of a group is the supremum of the (Pr\"{u}fer) ranks of its finite quotients, and for a prime $p$, the `upper $p$-rank' is the supremum of the sectional $p$-ranks of those quotients. The former is finite if and only if the…

We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We…

Given a finite group scheme $\cG$ over an algebraically closed field $k$ of characteristic $\Char(k)=p>0$, we introduce new invariants for a $\cG$-module $M$ by associating certain morphisms $\deg^j_M : U_M \lra \Gr_d(M) \ \…

We introduce a family of mathematical objects called $\mathcal{P}$-schemes, where $\mathcal{P}$ is a poset of subgroups of a finite group $G$. A $\mathcal{P}$-scheme is a collection of partitions of the right coset spaces $H\backslash G$,…

We investigate the representation theory of domestic group schemes $\mathcal{G}$ over an algebraically closed field of characteristic $p > 2$. We present results about filtrations of induced modules, actions on support varieties, Clifford…

Let $p$ be a prime. Given a split semisimple group scheme $G$ over a normal integral domain $R$ which is a faithfully flat $\mathbb Z_{(p)}$-algebra, we classify all finite dimensional representations $V$ of the fiber $G_K$ of $G$ over…

We prove that if a finite group scheme $G$ over a field $k$ has essential dimension one, then it embeds in $PGL_{2/k}$. We use this to give an explicit classification of all infinitesimal group schemes of essential dimension one over any…

The rank rk(G) of a profinite group G is the supremum of d(H), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure…

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places…

We consider the structure of finite $p$-groups $G$ having precisely three characteristic subgroups, namely $1$, $\Phi(G)$ and $G$. The structure of $G$ varies markedly depending on whether $G$ has exponent $p$ or $p^2$, and, in both cases,…

We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups, and obtains a…

Let $\mathscr{B}_0(\mathcal{G})\subseteq k\mathcal{G}$ be the principal block algebra of the group algebra $k\mathcal{G}$ of an infinitesimal group scheme $\mathcal{G}$ over an algebraically closed field $k$ of characteristic ${\rm…

Many open conjectures in the representation theory of finite groups can be studied by reducing them to related questions about quasi-simple groups. In such studies, $p$-radical subgroups typically play a critical role. To classify the…

Let $G$ be a finite group, and let $\mathbf{K}_p$ denote the completion at $p$ of the complex $K$-theory spectrum. $\mathbf{K}_p$ is a commutative ring spectrum that in some ways is very similar to the usual ring $\mathbf{Z}_p$ of $p$-adic…

Let $R$ be a finite unital commutative ring. We introduce a new class of finite groups, which we call hereditary groups over $R$. Our main result states that if $G$ is a hereditary group over $R$ then a unital algebra isomorphism between…