Maximal non-commuting set in 2-generated p-groups of class two

Let G be a group. A set S in G is said to be non-commuting if $$xy \ne yx$$ for any two distinct elements $$x,y \in S$$ . We define w(G) to be the maximum possible cardinality of a non-commuting set in G. In this paper, we determine w(G) for a finite non-abelian p-group G such that G/Z(G) is metacyclic by obtaining an abelian centralizers cover of this group. As a consequence, we show that the set of all commuting automorphisms of a finite non-abelian p-group G such that G/Z(G) is metacyclic, forms a subgroup of $$\mathrm {Aut}(G)$$ .

This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class C including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class C is both finite-of-exponent-e(n)–by–nilpotent-of-class≤c(n) and nilpotent-of-class≤c(n)–by–finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.

Given an additive group G and nonempty subsets S, T of G, let S+T denote the set ﹛s + t | s ∊ S, t ∊ T﹜, S the complement of S in G and |S| the cardinality of S. We call S a sum-free set in G if (S+S) ⊆ S. If, in addition, |S| ≥ |T| for every sum-free set T in G, then we call S a maximal sum-free set in G. We denote by λ(G) the cardinality of a maximal sum-free set in G.

Extract Although examples drive this book, we first provide a whirlwind survey of the general concepts of operator theory. We do not expect the student to master these topics now since they are covered in future chapters. A Hilbert space H is a complex vector space endowed with an inner product ⟨x,y⟩ that defines a norm ‖x‖=⟨x,x⟩ with respect to which H is (Cauchy) complete. The inner product on a Hilbert space satisfies the Cauchy–Schwarz inequality |⟨x,y⟩|⩽‖x‖‖y‖ for all x,y∈H⁠. Examples of Hilbert spaces include Cn (complex Euclidean space), ℓ2 (the space of square-summable complex sequences), and L2[0,1] (the Lebesgue space of square-integrable, complex-valued functions on [0,1]⁠). Vectors x,y in a Hilbert space H are orthogonal if ⟨x,y⟩=0⁠. The dimension of a Hilbert space H is the cardinality of a maximal set of nonzero orthogonal vectors. This book is almost exclusively concerned with Hilbert spaces of countable dimension. Every such Hilbert space has an orthonormal basis (un)n=1∞⁠, a (possibly finite) maximal orthogonal set such that ⟨um,un⟩=δmn for all m,n⩾1⁠. With respect to an orthonormal basis (un)n=1∞⁠, each x∈H enjoys a generalized Fourier expansion x=∑n=1∞⟨x,un⟩un that satisfies Parseval’s identity ‖x‖2=∑n=1∞|⟨x,un⟩|2. A subspace (a norm-closed linear submanifold) of H is itself a Hilbert space with the operations inherited from H⁠. If (wn)n=1∞ is a (possibly finite) orthonormal basis for a subspace M of H and x∈H⁠, then PMx=∑n=1∞⟨x,wn⟩wn belongs to M and satisfies ‖x−PMx‖⩽‖x−y‖ for every y∈M⁠. In short, PMx is the unique closest vector to x in M⁠. Furthermore, PM defines a linear transformation on H whose range is M⁠. It is called the orthogonal projection of H onto M and it satisfies PM2=PM and ⟨PMx,y⟩=⟨x,PMy⟩ for all x,y∈H⁠. Let H and K be Hilbert spaces. A linear transformation A:H→K is bounded if ‖A‖=sup{‖Ax‖K:‖x‖H=1} is finite. Let B(H,K) denote the set of bounded linear operators from H to K⁠. We write B(H) for B(H,H)⁠. The quantity ‖A‖ is the norm of A. Since B(H) is closed under addition and scalar multiplication, it is a vector space. Furthermore, since ‖A+B‖⩽‖A‖+‖B‖ and ‖cA‖=|c|‖A‖ for all A,B∈B(H) and c∈C⁠, it follows that B(H) is a normed vector space. Endowed with this norm, B(H) is complete and thus forms a Banach space. Moreover, the composition AB belongs to B(H) and ‖AB‖⩽‖A‖‖B‖ for all A,B∈B(H)⁠. Therefore, B(H) is a Banach algebra. For each A∈B(H,K)⁠, there is a unique A*∈B(K,H) such that ⟨Ax,y⟩K=⟨x,A*y⟩H for all x∈H and y∈K⁠. The operator A* is the adjoint of A; it is the analogue of the conjugate transpose of a matrix. One can show that A↦A* is conjugate linear, that A**=A⁠, ‖A‖=‖A*‖⁠, and ‖A*A‖=‖A‖2⁠. This additional structure upgrades B(H) from a Banach algebra to a C*-algebra. One can exploit adjoints to obtain information about the kernel and range of an operator. For most of the operators A∈B(H) covered in this book, we give the matrix representation [A]=[⟨Auj,ui⟩]i,j=1∞ with respect to an orthonormal basis (un)n=1∞ for H⁠. This matrix representation [A] defines a bounded operator x↦[A]x on the Hilbert space ℓ2 of square summable sequences that is structurally identical to A. Schur’s theorem helps us determine which infinite matrices define bounded operators on ℓ2⁠. Many of the operators covered in this book, such as the Cesàro operator, the Volterra operator, weighted shifts, Toeplitz operators, and Hankel operators, have fascinating structured-matrix representations. An important class of operators is the compact operators. These are the A∈B(H) such that (Axn)n=1∞ has a convergent subsequence whenever (xn)n=1∞ is a bounded sequence in H⁠. Equivalently, an operator is compact if it takes each bounded set to one whose closure is compact. Each finite-rank operator is compact and every compact operator is the norm limit of finite-rank operators. The compact operators form a norm-closed, *-closed ideal within B(H)⁠. Some operators have a particularly close relationship with their adjoint. For example, the operator M on L2[0,1] defined by (Mf)(x)=xf(x) satisfies M*=M⁠. Such operators are selfadjoint. If µ is a positive finite compactly supported Borel measure on C⁠, then the operator N on L2(μ) defined by (Nf)(z)=zf(z) satisfies (N*f)(z)=z―f(z)⁠, and thus N*N=NN*⁠. Such operators are normal. The operator (Uf)(eiθ)=eiθf(eiθ) on L2(T) satisfies U*U=UU*=I⁠. Such operators are unitary. Unitary operators preserve the ambient structure of Hilbert spaces and can serve as a vehicle to relate A∈B(H) with B∈B(K)⁠. We say that A is unitarily equivalent to B if there is a unitary U∈B(H,K) such that UAU*=B⁠. Unitary equivalence is often used to identify seemingly complicated operators with relatively simple ones.

Let p be a prime and K be a number field with non-trivial p-class group Cl p K . A crucial step in identifying the Galois group G p G K = of the maximal unramified pro-p extension of K is to determine its two-stage approximation 2 G p K = M , that is the second derived quotient G G M . The family 1 K of abelian type invariants of the p -class groups Cl p L of all unramified cyclic extensions L K of degree p is called the indexp abelianization data (IPAD) of K . It is able to specify a finite batch of contestants for the second p -class group M of K . In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD ( ) 1 2 , K K includes data on unramified abelian extensions L K of degree 2 p and enables sharper bounds for the order of M in the case ( ) Cl , , p K p p p , where current implementations of the p-group generation algorithm fail to produce explicit contestants for M , due to memory limitations. The iterated IPAD of second order ( ) 2 K contains information on non-abelian unramified extensions L/K of degree p 2 , or even p 3 , and admits the identification of the p-class tower group G for various infinite series of quadratic fields ( ) K d = with ( ) Cl , p K p p possessing a p-class field tower of exact length 3 p K = as a striking novelty.

A finite group G is said to be a minimal non-[Formula: see text] group if G is not a group of class ≤ n whose proper subgroups are of class ≤ n. In this paper, we give a complete classification of p-groups H of odd order with d(H) = 2 and c(H) = 2. Based on the classification of H, minimal non-[Formula: see text]p-groups G are classified for p > 3. If p > 3, then we have G3 ≅ Cp or G3 ≅ Cp × Cp. In this paper, we deal with the case when G3 ≅ Cp. In another paper [A classification of finite p-groups whose proper subgroups are of class ≤ 2(II), accepted] we deal with the case when G3 ≅ Cp × Cp.

Let p≥2 be a given prime number. We prove, for any number field kappa and any integer e≥1, the p-rank e-conjecture, on the p-class groups Cl_F, for the family F_kappa^p^e of towers F/kappa built as successive degree p cyclic extensions (without any other Galois conditions) such that F/kappa be of degree p^e, namely: #(Cl_F[p])<<_{kappa,p^e,e}(√D_F)^e, where D_F is the absolute value of the discriminant (Theorem 3.6) and, more generally, #(Cl_F[p^r])<<_{kappa,p^e,e}(√D_F)^e, for any r≥1 fixed. This Note generalizes the case of the family F_Q^p (Genus theory and e-conjectures on p-class groups, J. Number Theory 207, 423--459 (2020)), whose techniques appear to be ``universal'' for all relative degree p cyclic extensions and use the Montgomery--Vaughan result on prime numbers. Then we prove, for F_kappa^p^e, the p-rank e-conjecture on the cohomology groups H^2(G_F,Z_p) of Galois p-ramification theory over F (Theorem 4.3) and for some other classical finite p-invariants of F, as the Hilbert kernels and the logarithmic class groups.

Let r be a positive rational integer, and G a finite group. A subgroup H of G is called a r minimal (r-maximal) subgroup if there exists a subgroup chain 1 = H o ~< H 1 ~< -.. ~<H, = H(H = Ho ~< Hx ~<'" ~< H, = G), where H i is maximal in Hi+ x, we call Han S-quasinormal subgroup of G, if HP = PH for any Sylow subgroup P of G. If H is S-quasinormal in G, we write H SQN G. A subgroup H of G is called a quasinormal subgroup if HK = KH for any subgroup K of G. We write H QN G when H is a quasinormal subgroup of G. G is said to be an SQN-r-group(QN-r-group, PN-r-group) if every r-minimal subgroup of G is S-quasinormal(quasinormal, normal) in G. Clearly, a PN-r-group is a ON-r-group, and a QN-r-group is an SQN-r-group. Gaschfitz and Ito proved that PNl -g roup G is solvable and its commutator subgroup G' isp-nilpotent for any odd prime p. Hence the Fitting length Fl(G) ~< 3 [5, th5.7 p436]. Buckley proved in [1] that PN-1group of odd order is supersolvable. Sastry and Deskins had studied in I'3] PN-r-group and QNr-group. For instance, they proved that QN-r-group (r = 1,2,3) is solvable and QN-l-group and QN-2-group have Fitting length at most 4, the Fitting length of QN-3-group is not more than 5. This paper will study the SQN-r-groups. In section 2, we explore the properties of S--quasinormal subgroup and prove that the normal Hall subgroups of an S-quasinormal subgroup of a finite group is also an S-quasinormal subgroup. In section 3, we prove that SQN-l-group is solvable and the nitpotent residual is p-nilpotent for any odd prime p, so the Fitting length is at most 3. In section 4, we prove that SQN-2-group and SQN-3-group are either supersolvable or isomorphic to the several classes groups given in section t. Moreover, the commutator subgroups are nilpotent, so the Fitting lengths are at most 2. In section 5, we show that SQN-4-group and SQN-5--group are either solvable and the Fitting lengths at most 3 or non-solvable with only one non-abelian principal factor ~r is isomorphic ~o the simple group PSL(2,p). Groups in this .paper are all of finite order. The author wishes to thank his supervisor Professor Duan Xuefu and-Xu Mingyiao for their encouragement and many instructions.

Let k be an algebraic number field (of finite degree) and K/k be a (finite) Galois extension with Galois group R. In connection with the class field theory, Weil [101 derived a certain canonical factor set a(o, r) of 9 in the idele-class group (K of K, which is uniquely determined by K/k in the sense of equivalence. For instance, a (mod R') -?f t r(o, A) (Ek k, essentially) (mod NK/k( RK)) is the Artin reciprocity law isomoprhism of 9/9' and Sk/NK/k( SK), where 9' is the commutator subgroup of R. A different approach to this (idele-class) factor set was given in [6]. Let, for each pair o, r, a(o, r) be a representative idele of a(o, r). Then (3a)(p, o, r) = a(o, r)a(po, r)-'a(p, o-r)a(p, oY)is, for each triple p, a, r, a principal idele, i.e. a non-zero element of K. We obtain thus a 3-cochain a(p, a, r) = (6a)(p, a, r) of 9 in the principal idele group PK of K, which is in fact a 3-cocycle. We saw, in [6], that this 3-cycocle a, or more precisely its cohomology class, attached in invariant fashion to the Galois extension K/k, has the order m/m', where m = (K: k) and m' is the least common multiple of the p-degrees of K/k, p running over all primes in k. It was shown further, in a joint paper [4] by Hochschild and the writer, that the cyclic group, of order m/m', generated by this 3-cohomology class is exactly the group of all 3-cohomology classes of 9 in PK with ceiling. This last group is in turn, according to EilenbergMacLane [1], nothing but the group of all Teichmtiller 3-cohomology classes for K/k, belonging to central simple algebras over K to which every element of ft can be extended (as automorphism). Now arises the problem to determine the exact algebra-class (though not unique) which possesses our a, attached to K/k, as its Teichmiiller-class. The present work is devoted to this problem and our solution is given in our Theorem in ?3. If a central simple algebra A over K, to which every automorphism of K/k can be extended, is given in a crossed-product form for instance, we can obtain its Teichmiiller-cocycle in a more or less explicit process involving the HilbertSpeiser theorem. On the other hand, in order to obtain our 3-cocycle a, we repeat the construction of the canonical idele-class factor set a given in [6], which is nothing but the global analogy of the local construction in Hochschild [3]1 and makes use of the idele interpretation of the principal genus theorem of Noether, but with more detailed analysis. Namely, we carry out all the processes in terms of ideles, rather than of idele-classes, preserving thus all principal idele factors which were discarded in [6]. This construction, involving the theorem of everywhere splitting algebras, the Hilbert-Speiser theorem and its idble analogy, leads to a. Though fairly explicit, these constructions of a and the Teichmfiller-

Independence polynomials of well-covered graphs: Generic counterexamples for the unimodality conjecture

For a given \({\epsilon > 0}\) , we show that there exist two finite index subgroups of \({PSL_2(\mathbb{Z})}\) which are \(({1+\epsilon})\) -quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any \({\epsilon > 0}\) there are two finite regular covers of the Modular once punctured torus T 0 (or just the Modular torus) and a \({(1+\epsilon)}\) -quasiconformal map between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S p) of the punctured solenoid S p under the action of the corresponding Modular group (which is the mapping class group of S p [6], [7]) has the closure in T(S p) strictly larger than the orbit and that the closure is necessarily uncountable.

Identifying the extremal (minimal and maximal) sets from a collection of sets is an important subproblem in the areas of data-mining and satisfiability checking. For example, extremal set finding algorithms are used in the context of mining maximal frequent itemsets, and for simplifying large propositional satisfiability instances derived from real world tasks such as circuit routing and verification. In this paper, we describe two new algorithms for the task and detail their performance on real and synthetic data. Each algorithm leverages an entirely different principle – one primarily exploits set cardinality constraints, the other lexicographic constraints. Despite the inherent difficulty of this problem (the best known worst-case bounds are nearly quadratic), we show that both these algorithms provide excellent performance in practice, and can identify all extremal sets from multi-gigabyte itemset data using only a single processor core. Both algorithms are concise and can be implemented in no more than a few hundred lines of code. Our reference C++ implementations are open source and available for download.

The class of effectively closed infinite-genus surfaces, defining the completion of the domain of string perturbation theory, can be included in the category OG, which is characterized by the vanishing capacity of the ideal boundary. The cardinality of the maximal set of end points is shown to be 2N. The product of the coefficient of the genus-g superstring amplitude in four dimensions by 2g in the g→∞ limit is an exponential function of the genus with a base comparable in magnitude to the unified gauge coupling. The value of the string coupling is consistent with the characteristics of configurations which provide a dominant contribution to a finite vacuum amplitude.

It is known that a (generalized) soluble group whose proper subgroups of infinite rank are abelian either is abelian or has finite rank. It is proved here that if G is a group of infinite rank such that all its proper subgroups of infinite rank have locally finite commutator subgroup, then the commutator subgroup G' of G is locally finite, provided that G satisfies a suitable generalized solubility condition. Moreover, a similar result is obtained for groups whose proper subgroups of infinite rank are quasihamiltonian.

The root number class and Chinburg's second invariant