Iulian I. Simion

Groups equal to a product of three conjugate subgroups

Let G be a finite non-solvable group. We prove that there exists a proper subgroup A of G such that G is the product of three conjugates of A, thus replacing an earlier upper bound of 36 with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group G with a BN-pair and a finite Weyl group W satisfies

Minimal length factorizations of finite simple groups of Lie type by unipotent Sylow subgroups

G = {\left( {B{n_0}B} \right)^2} = B{B^{{n_0}}}B

On refinements of the Bruhat decomposition

where n0 is any preimage of the longest element of W. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of B in G. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.

Centers of Centralizers of Unipotent Elements in Exceptional Algebraic Groups

We consider factorizations of a finite group G into conjugate subgroups, G = Ax1⋯Axk for A ≤ G and x1,…,xk ∈ G, where A is nilpotent or solvable. We derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group which, for a large class of groups, is linear in the non-solvable length of G. We also show that every solvable group G is a product of at most 1 + clog |G : C| conjugates of a Carter subgroup C of G, where c is a positive real constant. Finally, using these results we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.

Double centralizers of unipotent elements in simple algebraic groups of type G2, F4 and E6

Abstract We prove that every finite simple group G of Lie type satisfies G = UU-UU-, where U is a unipotent Sylow subgroup of G and U- is its opposite. We also characterize the cases for which G = UU-U. These results are best possible in terms of the number of conjugates of U in the above factorizations.

Primitive permutation groups as products of point stabilizers

A conjecture of De Concini Kac and Procesi provides a bound on the minimal possible dimension of an irreducible module for quantized enveloping algebras at an odd root of unity. We pose the problem of the existence of modules whose dimension equals this bound. We show that this question cannot have a positive answer in full generality and discuss variants of this question.