Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.     

Invariants of infinite groups generated by reflections relative to lines

We distinguish nine rings of invariants of infinite groups generated by oblique reflections relative to lines of Euclidean space, and prove that a ring of invariants of any infinite group generated by such reflections is contained in one of these nine rings.Translated from Ukrainskii Geometricheski Sbornik, No. 33, pp. 65–69, 1990.     

Invariants of linear groups generated by matrices with two nonunit eigenvalues

A theorem on the structure of the algebra of invariants of the commutant of a group generated by pseudoreflections is improved. In particular, it is shown that this algebra is a complete intersection. A series of counterexamples to Stanley's conjecture is constructed in dimension 4. Results supporting this conjecture for primitive groups of large dimension are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 120–130, 1982.     

Tight frames generated by finite nonabelian groups

Let $\cal H$ be a Hilbert space of finite dimension d, such as the finite signals ? ₂(d) or a space of multivariate orthogonal polynomials, and n?≥?d. There is a finite number of tight frames of n vectors for $\cal H$ which can be obtained as the orbit of a single vector under the unitary action of an abelian group G (of symmetries of the frame). Each of these so called harmonic frames or geometrically uniform frames can be obtained from the character table of G in a simple way. These frames are used in signal processing and information theory. For a nonabelian group G there are in general uncountably many inequivalent tight frames of n vectors for $\cal H$ which can be obtained as such a G-orbit. However, by adding an additional natural symmetry condition (which automatically holds if G is abelian), we obtain a finite class of such frames which can be constructed from the character table of G in a similar fashion to the harmonic frames. This is done by identifying each G-orbit with an element of the group algebra ?G (via its Gramian), imposing the condition in the group algebra, and then describing the corresponding class of tight frames.     

Subgroups generated by small classes in finite groups

Let be the subgroup of generated by all elements that lie in conjugacy classes of the two smallest sizes. Avinoam Mann showed that if is nilpotent, then has nilpotence class at most . Using a slight variation on Mann's methods, we obtain results that do not require us to assume that is nilpotent. We show that if is supersolvable, then is nilpotent with class at most , and in general, the Fitting subgroup of has class at most .

     

Invariants of finite linear groups on relatively free algebras

It is shown that if G is a finite group of degree preserving automorphisms of R, the ring of n×n generic matrices over a field of characteristic zero generated by d > 1 elements, then the fixed ring R^G can never be generated by d elements unless n = 1 and G is a quasireflection group. As a consequence, for n > 1, R^G is never a generic matrix ring.     

Polycyclic groups with modular finite homomorphic images

A group G is said to be a modular group if it has modular subgroup lattice. We will prove in this paper that a polycyclic group G is modular if and only if all its finite homomorphic images are modular groups. Similar results will also be obtained for other conditions of modular type.     

The finite images of finitely generated groups

Given any sequence of non-abelian finite simple primitive permutationgroups S_n, we construct a finitely generated group G whose profinitecompletion is the infinite permutational wreath product ...S_n S_n–1 ... S₀. It follows that the upper compositionfactors of G are exactly the groups S_n. By suitably choosingthe sequence S_n we can arrange that G has any one of a continuousrange of slow, non-polynomial subgroup growth types. We alsoconstruct a 61-generator perfect group that has every non-abelianfinite simple group as a quotient. 2000 Mathematics SubjectClassification: 20E07, 20E08, 20E18, 20E32.