Monomorphism categories associated with symmetric groups and parity in finite groups

Monomorphism categories of the symmetric and alternating groups are studied via Cayley’s Em-bedding Theorem. It is shown that the parity is well defined in such categories. As an application, the parity in a finite group G is classified. It is proved that any element in a group of odd order is always even and such a group can be embedded into some alternating group instead of some symmetric group in the Cayley’s theorem. It is also proved that the parity in an abelian group of even order is always balanced and the parity in an nonabelian group is independent of its order.     

Special unipotent groups are split

We show that over any field k, a smooth unipotent algebraic k-group is special if and only if it is k-split.     

Witt groups and unipotent elements in algebraic groups

Let G be a semisimple algebraic group defined over an algebraicallyclosed field K of good characteristic p>0. Let u be a unipotentelement of G of order p^t, for some t N. In this paper it isshown that u lies in a closed subgroup of G isomorphic to theit Witt group W^t(K), which is a t-dimensional connected abelianunipotent algebraic group. 2000 Mathematics Subject Classification:20G15.     

On exact actions of unipotent groups

Any affine variety with a d-exact action of a unipotent group can be embedded in an affine space preserving d-exactness. Furthermore, we can find such an ambient space which has some other good properties. The key idea of the proof is describing the property “d-exact” by means of inequalities.     

Constructing arithmetic subgroups of unipotent groups

Let G be a unipotent algebraic subgroup of some defined over . We describe an algorithm for finding a finite set of generators of the subgroup . This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.     

Length-type parameters of finite groups with almost unipotent automorphisms

Let α be an automorphism of a finite group G. For a positive integer n, let E _G,n (α) be the subgroup generated by all commutators [...[[x,α],α],…,α] in the semidirect product G 〈α〉 over x ∈ G, where α is repeated n times. By Baer’s theorem, if E _G,n (α)=1, then the commutator subgroup [G,α] is nilpotent. We generalize this theorem in terms of certain length parameters of E _G,n (α). For soluble G we prove that if, for some n, the Fitting height of E _G,n (α) is equal to k, then the Fitting height of [G,α] is at most k + 1. For nonsoluble G the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F _h* (H) = H, where F ₀* (H) = 1, and F _i+1* (H) is the inverse image of the generalized Fitting subgroup F*(H/F _i *(H)). Let m be the number of prime factors of the order |α| counting multiplicities. It is proved that if, for some n, the generalized Fitting height E _G,n (α) of is equal to k, then the generalized Fitting height of [G,α] is bounded in terms of k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λE _G,n (α)= k, then the nonsoluble length of [G,α] is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

Weakly closed unipotent subgroups in Chevalley groups

The goal of this note is to classify the weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that the Lie algebra of a connected simple algebraic group fails to be a so-called 2F-module.     

Cohomologically trivial internal categories in categories of groups with operations

We describe cohomologically trivial internal categories in the categoryC of groups with operations satisfying certain conditions ([15], [16]). As particular cases we obtain: ifC=Gr, H⁰(C, –)=0 iff C is a connected internal category; ifC=Ab,H ¹(C, –)=0 iff C is equivalent to the discrete internal category (Cokerd, Cokerd, 1, 1, 1, 1). We also discuss related questions concerning extensions, internal categories, their cohomology and equivalence in the categoryC.     

Modular categories from finite crossed modules

It is known that finite crossed modules provide premodular tensor categories. These categories are in fact modularizable. We construct the modularization and show that it is equivalent to the module category of a finite Drinfeld double.     

Normal,unipotent subgroup schemes of reductive groups

Let G be a reductive group over a field k of characteristic p. Let $k^{\mathrm{sep}}$ be a separable closure of k. If $p\ne 2$, there exists a linear representation of G that is faithful and semisimple; moreover, any unipotent, normal subgroup scheme of G is trivial. For $p=2$, these two properties hold if and only if $G_{k^{\mathrm{sep}}}$ has no direct factor that is isomorphic to ${\mathbf{SO}}_{2n+1}$ for some $n?1$. To cite this article: A. Vasiu, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Some small unipotent representations of indefinite orthogonal groups

We study a family of small unitary representations of indefinite orthogonal groups. These representations arise as analytic continuations of the discrete series and were studied extensively by Knapp in [K3]. We complete Knapp's analysis by proving that they are irreducible. In order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one. Moreover, we prove that the K-type structure of each representation matches (up to a shift) the K-type structure of the ring of functions on the closure a nilpotent orbit on .     

Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group G, a parabolic subgroup P _J , and its unipotent radical U _J , we determine the ring structure of the cohomology ring H^?((U _J )₁, k). We also obtain new results on computing H^?((P _J )₁, L(??)) as an L _J -module where L(??) is a simple G-module with highest weight ?? in the closure of the bottom p-alcove. Finally, we provide generalizations of all our results to small quantum groups at a root of unity.