Finite Matrix Groups Research Articles

Finite quaternionic matrix groups

Let D \mathcal {D} be a definite quaternion algebra such that its center has degree d d over Q \mathbb {Q} . A subgroup G G of G L n ( D ) GL_n(\mathcal {D}) is absolutely irreducible if the Q \mathbb {Q} -algebra spanned by the matrices in G G is D n × n \mathcal {D}^{n\times n} . The finite absolutely irreducible subgroups of G L n ( D ) GL_n(\mathcal {D}) are classified for n d ≤ 10 nd \leq 10 by constructing representatives of the conjugacy classes of the maximal finite ones. Methods to construct the groups and to deal with the quaternion algebras are developed. The investigation of the invariant rational lattices yields quaternionic structures for many interesting lattices.

Finite Matrix Groups over Nilpotent Group Rings

We study groups of matricesSGLn(ZΓ) of augmentation one over the integral group ring ZΓ of a nilpotent group Γ. We relate the torsion ofSGLn(ZΓ) to the torsion of Γ. We prove that all abelianp-subgroups ofSGLn(ZΓ) can be stably diagonalized. Also, all finite subgroups ofSGLn(ZΓ) can be embedded into the diagonal Γn<SGLn(ZΓ). We apply matrix results to show that if Γ is nilpotent-by-(Π′-finite) then all finite Π-groups of normalized units in ZΓ can be embedded into Γ.

Finite rational matrix groups

Finite rational matrix groups

Zeros of equivariant vector fields: Algorithms for an invariant approach

We present a symbolic algorithm to solve for the zeros of a polynomial vector field equivariant with respect to a finite subgroup of O (n). We prove that the module of equivariant. polynomial maps for a finite matrix group is Cohen-Macaulay and give an algorithm to compute a fundamental basis. Equivariant normal forms are easily computed from this basis. We use this basis to transform the problem of finding the zeros of an equivariant map to the problem of finding zeros of a set of invariant polynomials. Solving for the values of fundamental polynomial invariants at the zeros effectively reduces each group orbit of solutions to a single point. Our emphasis is on a computationally effective algorithm and we present our techniques applied to two examples.

Irreducible finite integral matrix groups of degree 8 and 10

The lattices of eight- and ten-dimensional Euclidean space with irreducible automorphism group or, equivalently, the conjugacy classes of these groups inGLn(Z)\mathrm {GL}_n(\mathbb {Z})forn=8,10n = 8,10, are classified in this paper. The number of types is 52 in the casen=8n = 8, and 47 in the casen=10n = 10. As a consequence of this classification one has 26, resp. 46, conjugacy classes of maximal finite irreducible subgroups ofGL8(Z)\mathrm {GL}_8(\mathbb {Z}), resp.GL10(Z)\mathrm {GL}_{10}(\mathbb {Z}). In particular, each such group is absolutely irreducible, and therefore each of the maximal finite groups of degree 8 turns up in earlier lists of classifications.

Galois theory on the line in nonzero characteristic

The author surveys Galois theory of function fields with non-zero caracteristic and its relation to the structure of finite permutation groups and matrix groups.

Computing in permutation and matrix groups III: Sylow subgroups

The ability of construct the Sylow subgroups of a large finite permutation or matrix group is fundamentalto a number of important algorithms for analyzing the abstract structure of such a group. In this paper we describe a backtrack algorithm which constructs a Sylow subgroup by successive cyclic extensions, starting with a cyclic subgroup of p -power order. The algorithm is capable of finding Sylow p -subgroups of order up to p 10 , for small primes p , in permutation groups having a degree of several hundred.