Maximal Abelian Subgroups Research Articles

Nilpotent Fuzzy Subgroups

In this paper, we introduce a new definition for nilpotent fuzzy subgroups, which is called the good nilpotent fuzzy subgroup or briefly g-nilpotent fuzzy subgroup. In fact, we prove that this definition is a good generalization of abstract nilpotent groups. For this, we show that a group G is nilpotent if and only if any fuzzy subgroup of G is a g-nilpotent fuzzy subgroup of G. In particular, we construct a nilpotent group via a g-nilpotent fuzzy subgroup. Finally, we characterize the elements of any maximal normal abelian subgroup by using a g-nilpotent fuzzy subgroup.

A Look at the Inner Structure of the 2-adic Ring $C^*$‡-algebra and Its Automorphism Groups

We undertake a systematic study of the so-called 2-adic ring C^* -algebra \mathcal Q_2 . This is the universal C^* -algebra generated by a unitary U and an isometry S_2 such that S_2U=U^2S_2 and S_2S_2^*+US_2S_2^*U^*=1 . Notably, it contains a copy of the Cuntz algebra \mathcal O_2=C^*(S_1, S_2) through the injective homomorphism mapping S_1 to US_2 . Among the main results, the relative commutant C^*(S_2)'\cap \mathcal Q_2 is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion \mathcal O_2\subset\mathcal Q_2 , namely the endomorphisms of \mathcal Q_2 that restrict to the identity on \mathcal O_2 are actually the identity on the whole \mathcal Q_2 . Moreover, there is no conditional expectation from \mathcal Q_2 onto \mathcal O_2 . As for the inner structure of \mathcal Q_2 , the diagonal subalgebra \mathcal D_2 and C^*(U) are both proved to be maximal abelian in \mathcal Q_2 . The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of \mathcal Q_2 . In particular, the semigroup of the endomorphisms fixing U turns out to be a maximal abelian subgroup of Aut (\mathcal Q_2) topologically isomorphic with C(\mathbb{T},\mathbb{T}) . Finally, it is shown by an explicit construction that Out (\mathcal Q_2) is uncountable and non-abelian.

The inner structure of boundary quotients of right LCM semigroups

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,\theta)$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in $2 \leq p<\infty$ generators to the $p$-adic ring $C^*$-algebra. For the case where $P$ is abelian and $C^*(G)$ is a maximal abelian subalgebra, we establish a picture for the automorphisms of the boundary quotient that fix $C^*(G)$ pointwise. This allows us to show that they form a maximal abelian subgroup of the entire automorphism group. The picture also leads to the surprising outcome that, for integral dynamics, every automorphism that fixes one of the natural Cuntz subalgebras pointwise is necessarily a gauge automorphism. Many of the automorphisms we consider are shown to be outer.

Maximal finite abelian subgroups of E8

The maximal finite abelian subgroups, up to conjugation, of the simple algebraic group of type E8 over an algebraically closed field of characteristic 0 are computed. This is equivalent to the determination of the fine gradings on the simple Lie algebra of type E8 with trivial neutral homogeneous component. The Brauer invariant of the irreducible modules for graded semisimple Lie algebras plays a key role.

Finite nonabelian p-groups of exponent &gt;p with a small number of maximal abelian subgroups of exponent &gt;p

Finite nonabelian p-groups of exponent &gt;p with a small number of maximal abelian subgroups of exponent &gt;p

On the number of minimal nonabelian subgroups in a nonabelian finite p-group

The number of minimal nonabelian subgroups in a nonabelian finite [Formula: see text]-group containing a maximal normal abelian subgroup of given index is estimated. Also the [Formula: see text]-group for which the above estimate is attained, are described in great detail.

Space-Time Defects and Group Momentum Space

We study massive and massless conical defects in Minkowski and de Sitter spaces in various space-time dimensions. The energy momentum of a defect, considered as an (extended) relativistic object, is completely characterized by the holonomy of the connection associated with its space-time metric. The possible holonomies are given by Lorentz group elements, which are rotations and null rotations for massive and massless defects, respectively. In particular, if we fix the direction of propagation of a massless defect in n+1-dimensional Minkowski space, then its space of holonomies is a maximal Abelian subgroup of the AN(n-1) group, which corresponds to the well known momentum space associated with the n-dimensional κ-Minkowski noncommutative space-time and κ-deformed Poincaré algebra. We also conjecture that massless defects in n-dimensional de Sitter space can be analogously characterized by holonomies belonging to the same subgroup. This shows how group-valued momenta related to four-dimensional deformations of relativistic symmetries can arise in the description of motion of space-time defects.

Characters of odd degree

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verication of the inductive McKay condition for groups of Lie type and primes ‘ such that a Sylow ‘-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specics of characters of odd degree, namely, that most of them lie in the principal Harish-Chandra series, then allow us to deduce from it the McKay conjecture for the prime 2, hence for characters of odd degree.

Maximal abelian subgroups of compact simple Lie groups of type E

We classify closed abelian subgroups of the automorphism group of a compact simple Lie algebra of type E whose centralizer has the same dimension as the dimension of the subgroup. These include maximal abelian subgroups and Jordan subgroups. We also describe Weyl groups of these abelian subgroups.

On recognizability of PSU3 (q) by the orders of maximal abelian subgroups

In [Li and Chen, A new characterization of the simple group A_1(p^n), Sib. Math. J., 2012], it is proved that the simple group A_1(p^n) is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as A_2(q), Monatsh. Math., 2013], the authors proved that if L=A_2(q), where q is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as L, is isomorphic to L or an extension of L by a subgroup of the outer automorphism group of L. In this paper, we prove that if L=PSU_3(q), where q is not a Fermat prime, then every finite group with the same orders of maximal abelian subgroups as L, is isomorphic to L or an extension of L by a field automorphism of L.

On CT and CSA groups and related ideas

Abstract A group is G commutative transitive or CT if commuting is transitive on nontrivial elements. A group G is CSA or conjugately separated abelian if maximal abelian subgroups are malnormal. These concepts have played a prominent role in the studies of fully residually free groups, limit groups and discriminating groups. They also play a role in the solution to the Tarski problems. CSA always implies CT however the class of CSA groups is a proper subclass of the class of CT groups. For limit groups and finitely generated elementary free groups they are equivalent. In this paper we examine the relationship between the two concepts. In particular, we show that a finite CSA group must be abelian. If G is CT, then we prove that G is not CSA if and only if G contains a nonabelian subgroup G 0 ${G_{0}}$ which contains a nontrivial abelian subgroup H that is normal in G 0 ${G_{0}}$ . For K a field the group PSL ⁢ ( 2 , K ) ${\mathrm{PSL}(2,K)}$ is never CSA but is CT if char ⁡ ( K ) = 2 ${\operatorname{char}(K)=2}$ and for fields K of characteristic 0 where -1 is not a sum of two squares in K. For characteristic p, for an odd prime p, PSL ⁢ ( 2 , K ) ${\mathrm{PSL}(2,K)}$ is never CT. Infinite CT groups G with a composition series and having no nontrivial normal abelian subgroup must be monolithic with monolith a simple nonabelian CT group. Further if a group G is monolithic with monolith N isomorphic to PSL ⁢ ( 2 , K ) ${\mathrm{PSL}(2,K)}$ for a field K of characteristic 2 and G is CT, then G ≅ N ${G\cong N}$ .

CommutativeC⁎-Algebras of Toeplitz Operators via the Moment Map on the Polydisk

We found that in the polydiskDnthere exist(n+1)(n+2)/2different classes of commutativeC⁎-algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that theC⁎-algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutativeC⁎-algebra of Toeplitz operators.

Commutative C*-Algebras Generated by Toeplitz Operators on the Super Unit Ball

We extend known results about commutative $C^*$-algebras generated Toeplitz operators over the unit ball to the supermanifold setup. This is obtained by constructing commutative $C^*$-algebras of super Toeplitz operators over the super ball $\mathbb{B}^{p|q}$ and the super Siegel domain $\mathbb{U}^{p|q}$ that naturally generalize the previous results for the unit ball and the Siegel domain. In particular, we obtain one such commutative $C^*$-algebra for each even maximal Abelian subgroup of automorphisms of the super ball.

A New Characterization of the Mathieu Groups and Alternating Groups with Degrees Primes

In the past thirty years, several kinds of quantitative characterizations of finite groups especially finite simple groups have been investigated by many mathematicians. Such as quantitative characterizations by group order and element orders, by element orders alone, by the set of sizes of conjugacy classes, by dimensions of irreducible characters, by the set of orders of maximal abelian subgroups and so on. Here the authors continue this topic in a new area tending to characterize finite simple groups with given orders by some special conjugacy class sizes, such as largest conjugacy class sizes, smallest conjugacy class sizes greater than 1 and so on.

Quantum error correction for a subgroup of the Pauli group using higher rank numerical range

Suppose noise is characterized by a quantum channel whose error operators are members of an arbitrary (not necessarily abelian) Pauli subgroup of -qubit Pauli group, where . In this paper, using a maximal abelian subgroup of , we obtained the largest , for which the joint rank- numerical range associated with members of is non-empty. It is proved that qubits codeword encodes data qubits. Moreover, we obtain recovery operators such that the decoded state automatically appears in the output with no syndrome measurements for such channels.

Confinement on R3 × S1: Continuum and lattice

There has been substantial progress in understanding confinement in a class of four-dimensional SU(N) gauge theories using semiclassical methods. These models have one or more compact directions, and much of the analysis is based on the physics of finite temperature gauge theories. The topology R3 × S1 has been most often studied using a small compactification circumference L such that the running coupling g2(L) is small. The gauge action is modified by a double-trace Polyakov loop deformation term, or by the addition of periodic adjoint fermions. The additional terms act to preserve Z(N) symmetry and thus confinement. An area law for Wilson loops is induced by a monopole condensate. In the continuum, the string tension can be computed analytically from topological effects. Lattice models display similar behavior, but the theoretical analysis of topological effects is based on Abelian lattice duality rather than on semiclassical arguments. In both cases, the key step is reducing the low-energy symmetry group from SU(N) to the maximal Abelian subgroup U(1)N-1 while maintaining Z(N) symmetry.

Groups with the same set of orders of maximal abelian subgroups

Let n &gt; 3 be an even number. In this paper, we show how the orders of maximal abelian subgroups of the finite group G can influence on the structure of G. More precisely, we show that if for a finite group G,M(G) = M(Bn(q)), then G ? Bn(q). Note that M(G) is the set of orders of maximal abelian subgroups of G. Let ?(G) denote the non-commuting graph of G. As a consequence of our result, we show that if G is a finite group with ?(G) ? (Bn(q)), then G ? Bn(q).

General Schlesinger Systems and Their Symmetry from the View Point of Twistor theory

Isomonodromic deformation of linear differential equations on ℙ1 with regular and irregular singular points is considered from the view point of twistor theory. We give explicit form of isomonodromic deformation using the maximal abelian subgroup H of G = GLN+1(ℂ) which appeared in the theory of general hypergeometric functions on a Grassmannian manifold. This formulation enables us to obtain a group of symmetry for the nonlinear system which is an Weyl group analogue NG (H)/H.

A Characterization of $$^2F_4(q)$$ 2 F 4 ( q ) by the set of orders of maximal Abelian subgroups

The simple group \(^2F_4(q)\) is proved to be uniquely determined by the set of orders of its maximal abelian subgroups, where \(q=2^{2m+1}\) and \(m\ge 1\).

Groups with the same orders of maximal abelian subgroups as $$A_2(q)$$

In Li and Chen (Sib. Math. J. 53(2), 243–247, 2012), it is proved that the simple group \(A_1(p^n)\) is uniquely determined by the set of orders of its maximal abelian subgroups. Let \(q=p^{\alpha }\) be a prime power and \(L=A_2(q)\). In this paper, we prove that if \(q\) is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as \(L\), is isomorphic to \(L\) or an extension of \(L\) by a subgroup of the outer automorphism group of \(L\).