Groups of Matrices That Act Monopotently

In the present article, the authors continue the line of inquiry started by Cigler and Jerman, who studied the separation of eigenvalues of a matrix under an action of a matrix group. The authors consider groups \Fam{G} of matrices of the form $\left[\small{\begin{smallmatrix} G & 0\\ 0& z \end{smallmatrix}}\right]$, where $z$ is a complex number, and the matrices $G$ form an irreducible subgroup of $\GL(\C)$. When \Fam{G} is not essentially finite, the authors prove that for each invertible $A$ the set $\Fam{G}A$ contains a matrix with more than one eigenvalue. The authors also consider groups $\Fam{G}$ of matrices of the form $\left[\small{\begin{smallmatrix} G & x\\ 0& 1 \end{smallmatrix}}\right]$, where the matrices $G$ comprise a bounded irreducible subgroup of $\GL(\C)$. When \Fam{G} is not finite, the authors prove that for each invertible $A$ the set $\Fam{G}A$ contains a matrix with more than one eigenvalue.

Translation planes of even order in which the dimension has only one odd factor

LetGbe an irreducible subgroup of the linear translation complement of a finite translation plane of orderqdwhereqis a power of2.GF(q)is in the kernel andd=2srwhereris an odd prime. A prime factor of|G|must divide(qd+1)∏i=1d(qi−1).One possibility (there are no known examples) is thatGhas a normal subgroupWwhich is aW-group for some primeW.The maximal normal subgroup0(G)satisfies one of the following:1. Cyclic. 2. Normal cyclic subgroup of indexrand the nonfixed-point-free elements in0(G)have orderr. 3.0(G)contains a groupWas above.

Solvable-by-finite subgroups of GL(2, F)

In a recent paper [5] Tits proves that a linear group over a field of characteristic zero is either solvable-by-finite or else contains a non-cyclic free subgroup. In this note we determine all the infinite irreducible solvable-by-finite subgroups of GL(2, F), where F is an algebraically closed field of characteristic zero. (Every reducible subgroup of GL(2, F) is metabelian.) In addition, we prove that an irreducible subgroup of GL(2, F) has an irreducible solvable-by-finite subgroup if and only if it contains an element of zero trace.