The exponents of nonabelian tensor products of groups

We prove that the exponent of the nonabelian tensor product of two locally finite groups can be bounded in terms of exponents of given groups. Several estimates for the exponents of nonabelian tensor squares are obtained. In particular, if the group G is nilpotent of class ≤3 and of finite exponent, then the exponent of its nonabelian tensor square divides the exponent of G.     

Groups of prime power order and their nonabelian tensor squares

We prove that the nonabelian tensor square of a powerful p-group is again a powerful p-group. Furthermore, If G is powerful, then the exponent of G ⊕ G divides the exponent of G. New bounds for the exponent, rank, and order of various homological functors of a given finite p-group are obtained. In particular, we improve the bound for the order of the Schur multiplier of a given finite p-group obtained by Lubotzky and Mann.     

Turing degrees of nonabelian groups

For a countable structure , the (Turing) degree spectrum of is the set of all Turing degrees of its isomorphic copies. If the degree spectrum of has the least degree , then we say that is the (Turing) degree of the isomorphism type of . So far, degrees of the isomorphism types have been studied for abelian and metabelian groups. Here, we focus on highly nonabelian groups. We show that there are various centerless groups whose isomorphism types have arbitrary Turing degrees. We also show that there are various centerless groups whose isomorphism types do not have Turing degrees.

     

Near integral domains on nonabelian groups

In this paper the concept of a special automorphism is introduced and used to analyze near integral domains having nonabelian addtive groups. We show that there are finite and infinite near integral domains having additive groups with arbitrary class of nilpotency. We also give another example of a non-nilpotent group which is the additive group of a near integral domain. Finally, nonabelian groups of order less than 1000 are examined to determine which can be the additive group of a near integral domain.Most of the results of this paper are contained in the author's doctoral dissertation at Boston University. The author thanks ProfessorD. W. Blackett for his guidance.     

Twisted Weyl groups of Lie groups and nonabelian cohomology

For a cyclic group A and a connected Lie group G with an A-module structure (with the additional assumptions that G is compact and the A-module structure on G is 1-semisimple if ), we define the twisted Weyl group W = W(G,A,T), which acts on T and H ¹(A,T), where T is a maximal compact torus of , the identity component of the group of invariants G ^A . We then prove that the natural map is a bijection, reducing the calculation of H ¹(A,G) to the calculation of the action of W on T. We also prove some properties of the twisted Weyl group W, one of which is that W is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.      

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.     

On Calabi-Yau threefolds with large nonabelian fundamental groups

In this short note we construct Calabi-Yau threefolds with nonabelian fundamental groups of order as quotients of the small resolutions of certain complete intersections of quadrics in that were first considered by M. Gross and S. Popescu.

     

Malliavin's theorem for weak synthesis on nonabelian groups

Malliavin's celebrated theorem on the failure of spectral synthesis for the Fourier algebra A(G) on nondiscrete abelian groups was strengthened to give failure of weak synthesis by Parthasarathy and Varma. We extend this to nonabelian groups by proving that weak synthesis holds for A(G) if and only if G is discrete. We give the injection theorem and the inverse projection theorem for weak X-spectral synthesis, as well as a condition for the union of two weak X-spectral sets to be weak X-spectral for an A(G)-submodule X of VN(G). Relations between weak X-synthesis in A(G) and A(G×G) and the Varopoulos algebra V(G) are explored. The concept of operator synthesis was introduced by Arveson. We extend several recent investigations on operator synthesis by defining and studying, for a V^∞(G)-submodule M of B(L²(G)), sets of weak M-operator synthesis. Relations between X-Ditkin sets and M-operator Ditkin sets and between weak X-spectral synthesis and weak M-operator synthesis are explored.     

Covering theorems for nonabelian simple groups. II

In this part II, I study the class C of an n-cycle or of an n ? 1-cycle in the alternating group A_n. For n = 4k ? 1, 4k, CCC covers A_n, but CC does not. For n = 4k + 1, CCC covers A_n; I do not know whether CC does. For n = 4k + 2, CC covers A_n [Part III concerns PSL(2, q).]     

Locally graded groups with complemented infinite nonabelian subgroups

Infinite nonabelian groups with complemented infinite nonabelian subgroups are investigated. It is proved that, under the condition of being locally graded, these groups are locally finite and solvable, and all nonabelian subgroups are complemented in them if and only if they are non-Chernikov groups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp, 1098–1100, July–August, 1991.     

Constructions of large translation nets with nonabelian translation groups

In this paper the first infinite series of translation nets with nonabelian translation groups and a large number of parallel classes are constructed. For that purpose we investigate partial congruence partitions (PCPs) with at least one normal component.Two series correspond to partial congruence partitions containing one normal elementary abelian component. The construction results by using some basic facts about the first cohomology group of the translation group G regarded as an extension of the normal component which itself is a group of central translations.The other series correspond to partial congruence partitions containing two normal nonabelian components. The constructions are based on the well known automorphism method which leads to so-called splitting translation nets. By investigating the Suzuki groups Sz(q), the protective unitary groups PSU(3, q ²) and the Ree groups R(q) as doubly transitive permutation groups, we obtain examples of nonabelian groups admitting a large number of pairwise orthogonal fixed-point-free group automorphisms.     

New difference sets in nonabelian groups of order 100

In two groups of order 100 new difference sets are constructed. The existence of a difference set in one of them has not been known. The correspondence between a (100, 45, 20) symmetric design having regular automorphism group and a difference set with the same parameters has been used for the construction. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 424–434, 2001     

Blocks of full defect with nonabelian metacyclic defect groups

Let p be an odd prime and let B be a p-block of a finite group G with a nonabelian metacyclic defect group P which is a Sylow p-subgroup of G. The purpose of this article is to study the ordinary and modular irreducible characters in B. In particular, we calculate k _i (B) and l _i (B) for an arbitrary nonnegative integer i.