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We define the concept of an [Formula: see text]-co-Hopfian abelian group, which is a nontrivial generalization of the classical notion of a co-Hopfian group. A systematic and comprehensive study of these groups is given in very different ways. Specifically, a representation theorem for [Formula: see text]-co-Hopfian groups is established as well as it is shown that there exists an [Formula: see text]-co-Hopfian group which is not co-Hopfian.

In this paper we prove that certain generalized free products of abelian subgroup separable groups, amalgamating an infinite cyclic subgroup, are abelian subgroup separable. Applying this, we derive that tree products of free groups or finitely generated nilpotent groups, amalgamating infinite cyclic subgroups, are abelian subgroup separable.

We prove that a group whose element orders divide 6 and 7 either is locally finite or an extension of a nontrivial elementary abelian 2-group by a group without involutions.

We analyse the composite functional equation $$f(x+2f(y))=f(x)+y+f(y)$$ on certain groups. In particular we give a description of solutions on abelian 3-groups and finitely generated free abelian groups. This is motivated by a work of Pal Burai, Attila Hazy and Tibor Juhasz, who described the solutions of the equation on uniquely 3-divisible abelian groups.

Let G be a finite nontrivial group with an irreducible complex character χ of degree d = χ(1). According to the orthogonality relation, the sum of the squared degrees of irreducible characters of G is the order of G. N. Snyder proved that, if G = d(d + e), then the order of the group G is bounded in terms of e for e > 1. Y. Berkovich demonstrated that, in the case e = 1, the group G is Frobenius with the complement of order d. This paper studies a finite nontrivial group G with an irreducible complex character Θ such that G ≤ 2Θ(1)2 and Θ(1) = pq where p and q are different primes. In this case, we have shown that G is a solvable group with an Abelian normal subgroup K of index pq. Using the classification of finite simple groups, we have established that the simple non-Abelian group, the order of which is divisible by the prime p and not greater than 2p 4 is isomorphic to one of the following groups: L 2(q), L 3(q), U 3(q), S z(8), A 7, M 11, and J 1.

In this paper, the maximal abelian dimension is algorithmically and computationally studied for the Lie algebra hn, of n×n upper-triangular matrices. More concretely, we define an algorithm to compute abelian subalgebras of hn besides programming its implementation with the symbolic computation package MAPLE. The algorithm returns a maximal abelian subalgebra of hn and, hence, its maximal abelian dimension. The order n of the matrices hn is the unique input needed to obtain these subalgebras. Finally, a computational study of the algorithm is presented and we explain and comment some suggestions and comments related to how it works.

For any finite abelian group A, we give the lower bound of ordp(|Hom(A, Sn)|), and determine the region of convergence of the p-adic power series 1 + Sigma(infinity)(n=1) vertical bar Hom(A, S-n)vertical bar X-n/ n!., 2000 Mathematics Subject Classication: Primary 05A15; Secondary 20B30; 20K01; 20K27

Let G be a group of squarefree exponent e which acts regularly on some abelian p-group V. If V is the union of a finite number of G-orbits and e divides p − 1, we show that G is the cyclic group of order e.

LetGbe an additive abelian group, letn⩾ 1 be an integer, letSbe a sequence overGof length |S| ⩾n+ 1, and let${\mathsf h}$(S) denote the maximum multiplicity of a term inS. Let Σn(S) denote the set consisting of all elements inGwhich can be expressed as the sum of terms from a subsequence ofShaving lengthn. In this paper, we prove that eitherng∈ Σn(S) for every termginSwhose multiplicity is at least${\mathsf h}$(S) − 1 or |Σn(S)| ⩾ min{n+ 1, |S| −n+ | supp (S)| − 1}, where |supp(S)| denotes the number of distinct terms that occur inS. WhenGis finite cyclic andn= |G|, this confirms a conjecture of Y. O. Hamidoune from 2003.

We give a complete description of Abelian groups that are totally P-stable for the following four natural types of subgroups: arbitrary subgroups, pure subgroups, elementary subsystems, and algebraically closed subgroups.

We study primary Abelian groups containing proper fully invariant subgroup isomorphic to the group. The admissable sequence of the Ulm–Kaplansky invariants for primary groups is used. Using these sequences, IF-groups are described in some important classes of Abelian p-group.

We estimate the order of an abelian p-subgroup A (p a prime) of a symmetric group of degree n, by a function of n and the rank k of A.

In this paper, we study the notion of T -noncosingular abelian groups, which means an abelian group whose nonzero endomorphisms are not small. We show that injective (divisible) and projective (free) groups are T -noncosingular. We proved that T -noncosingular torsion groups are exactly the direct sum of a semisimple group C and a divisible group D which does not have simple subgroups isomorphic to a subgroup of C. Many other interesting results are presented here, we also give some condition for torsion-free groups to be T noncosingular (Though torsion-free groups are K-noncosingular).

The notion of a generic theory GTh( $$ \mathcal{K} $$ , μ) with respect to a measure μ was introduced in [Algebra and Logic, 53, No. 6, 512-519 (2014)]. Here, generic theories for two series of cyclic groups are described based on elementary invariants for Abelian groups and using a measure generated by a Frechet filter. Axioms of generic theories are given, complete theories are characterized in terms of elementary invariants, and canonical models of complete theories are constructed.

Bieberbach groups are torsion free crystallographic groups. In this paper, our focus is on the Bieberbach groups with elementary abelian 2-group point group, The central subgroup of the nonabelian tensor square of a group is generated by for all in The purpose of this paper is to compute the central subgroups of the nonabelian tensor squares of two Bieberbach groups with elementary abelian 2-point group of dimension three.

In [6], givenany torsion-free abelian groups Gand H, the pure trace of Hin Gis *,:,GHHomfHfGHtr which is equivalent to the set ZnGHHomfHfngGgsomefor ,,:.The pure trace GHtr, is a pure fully invariant subgroup of G. A torsion-free abelian group Gis a trace group if for every pure fully invariant subgroup Mof G, .,GMtrMIn this paper, we giv further results of pure trace, trace group and characterize the direct sum of the trace groups of a torsion-free abelian groups.

In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.

The Birkhoff--Kakutani Theorem asserts that a topological group is metrizable if and only if it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups. We consider abelian groups whose topology is determined by a countable cofinal family of compact sets. These are the closed subgroups of Pontryagin--van Kampen duals of \emph{metrizable} abelian groups, or equivalently, complete abelian groups whose dual is metrizable. By investigating these connections, we show that also in these cases, the character can be estimated, and that it is determined by the weights of the \emph{compact} subsets of the group, or of quotients of the group by compact subgroups. It follows, for example, that the density and the local density of an abelian metrizable group determine the character of its dual group. Our main result applies to the more general case of closed subgroups of Pontryagin--van Kampen duals of abelian \v{C}ech-complete groups. In the special case of free abelian topological groups, our results extend a number of results of Nickolas and Tkachenko, which were proved using combinatorial methods. In order to obtain concrete estimations, we establish a natural bridge between the studied concepts and pcf theory, that allows the direct application of several major results from that theory. We include an introduction to these results and their use., Comment: Minor corrections. Final version before journal editing

Let F / k be a finite Galois extension of number fields with Galois group G, and A an abelian variety over k. We fix an odd prime p. When G is isomorphic to the dihedral group of order 4p, assuming the Birch and Swinnerton–Dyer conjecture and some technical assumptions, we prove that the Pontryagin dual of the (classical) Selmer group Sel p ( A F ) ∨ is annihilated by some element in the center of the group ring Z p [ G ] which is constructed from equivariant L-values. More generally, we study the relation between the annihilation of Selmer groups for non-abelian extensions and that for abelian extensions.

The Chermak–Delgado lattice of a finite group is a dual, modular sublattice of the subgroup lattice of the group. This paper considers groups with a quasi-antichain interval in the Chermak–Delgado lattice, ultimately proving that if there is a quasi-antichain interval between subgroups L and H with L ≤ H then there exists a prime p such that H/L is an elementary abelian p-group and the number of atoms in the quasi-antichain is one more than a power of p. In the case where the Chermak–Delgado lattice of the entire group is a quasi-antichain, the relationship between the number of abelian atoms and the prime p is examined; additionally, several examples of groups with a quasi-antichain Chermak–Delgado lattice are constructed.

Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on one hand, and abelian extensions and congruence solvability on the other hand. In particular, we show that a loop is congruence solvable (that is, an iterated abelian extension of commutative groups) if and only if it is not Boolean complete, reaffirming the connection between computational complexity and solvability. Finally, we briefly discuss relations between nilpotence and solvability for loops and the associated multiplication groups and inner mapping groups.

E-nilpotent and E-solvable Abelian groups are studied. The properties of such groups are studied, and examples illustrating the differences and connections between the investigated classes of groups are presented. The structure of E-solvable periodical, completely decomposable, coperiodic, split mixed, and other groups is shown.

LetG,Nbe a pair of groups whereGis a group andNis a normal subgroup ofG. Then the Schur multiplier of pairs of groupsG,Nis a functorial abelian groupMG,N. In this paper,MG,Nfor groups of orderp2qwherepandqare prime numbers are determined.

We say that a complex valued function defined on an Abelian group G is a local polynomial, if its restriction to every finitely generated subgroup of G is a polynomial. We prove that local spectral synthesis (that is, spectral synthesis using local polynomials instead of polynomials) holds on every Abelian group having countable torsion free rank. More precisely, there is a cardinal ω1≦κ≦2ω such that local spectral synthesis holds on an Abelian group G if and only if the torsion free rank of G is less than κ.

For every finite Abelian group and integer we construct a finitely presented group defined by explicit generators and relations such that its centre is isomorphic to .

We describe unary polynomial functions on nite groups G that are semidirect products of an elementary abelian group of exponent p and a cyclic group of prime order q, p ≠ q.

We prove the existence of a locally finite periodic part in every rank 1 Shunkov group with solvable finite subgroups, in every Shunkov group acting freely on an abelian group, and in the groups of affine transformations of neardomains with finite elements.

— Given an algebraic number field k and a finite group Γ, we write R(Ok[Γ]) for the subset of the locally free classgroup Cl(Ok[Γ]) consisting of the classes of rings of integers ON in tame Galois extensions N/k with Gal(N/k) ∼= Γ. We determine R(Ok[Γ]), and show it is a subgroup of Cl(Ok[Γ]) by means of a description using a Stickelberger ideal and properties of some cyclic codes, when k contains a root of unity of prime order p and Γ = V oC, where V is an elementary abelian group of order pr and C is a cyclic group of order m > 1 acting faithfully on V and making V into an irreducible Fp[C]-module. This extends and refines results of Byott, Greither and Sodaigui for p = 2 in Crelle, respectively of Bruche and Sodaigui for p > 2 in J. Number Theory, which cover only the case m = pr−1 and determine only the image R(M) of R(Ok[Γ]) under extension of scalars from Ok[Γ] to a maximal orderM⊃ Ok[Γ] in k[Γ]. The main result here thus generalizes the calculation of R(Ok[A4]) for the alternating group A4 of degree 4 (the case p = r = 2) given by Byott and Sodaigui in Compositio. Resume. — Etant donne un corps de nombres k et un groupe fini Γ, on note R(Ok[Γ]) le sous-ensemble du groupe de classes localement libre Cl(Ok[Γ]) forme par les classes d’anneaux d’entiers ON d’extensions galoisiennes moderees N/k avec Gal(N/k) ∼= Γ. Nous determinons R(Ok[Γ]), et montrons que c’est un sous-groupe de Cl(Ok[Γ]), au moyen d’une description utilisant un ideal de Stickelberger et des proprietes de certains codes cycliques, lorsque k contient une racine de l’unite d’ordre premier p et Γ = V o C, ou V est un groupe elementaire abelien d’ordre pr et C est un groupe cyclique d’ordre m > 1 agissant fidelement sur V et rendant V un Fp[C]-module irreductible. Ceci generalise et raffine des resultats de Byott, Greither et Sodaigui pour p = 2 dans Crelle, respectivement de Bruche et Sodaigui pour p > 2 dans J. Number Theory, lesquels couvrent seulement le cas m = pr−1 et determinent seulement l’image R(M) de R(Ok[Γ]) sous l’extension des scalaires de Ok[Γ] a un ordre maximalM⊃ Ok[Γ] dans k[Γ]. Le resultat principal ici generalise donc le calcul de R(Ok[A4]) pour le groupe alterne A4 de degre 4 (le cas p = r = 2) donne par Byott et Sodaigui dans Compositio.

A finitep-groupGis said to have the propertyP, if, for any abelian subgroupMofG, there is|MZ(G)/Z(G)|≤p. We show that ifGsatisfiesP, thenGhas the following two types: (1)Gis isoclinic to some stem groups of orderp5, which form an isoclinic family. (2)Gis isoclinic to a specialp-group of exponentp. Elementary structures of groups withPare determined.

We introduce a notion of compatible quasi-ordered groups which unifies valued and ordered abelian groups. It was proved by S.M. Fakhruddin that a compatible quasi-order on a field is always either an order or a valuation. We show here that the group case is more complicated than the field case and describe the general structure of a compatible quasi-ordered abelian group. We then define a notion of Hahn product of compatible quasi-ordered groups and generalize Hahn's embedding theorem to quasi-ordered groups. We also develop a notion of quasi-order-minimality and establish a connection with C-minimality, thus answering a question of F. Delon. Finally, we use compatible quasi-ordered groups to give an example of a C-minimal group which is neither an ordered nor a valued group.

We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero., Comment: 11 pages

Let X be a finite Abelian group, let ξ i , i = 1, 2,…, n, n ≥ 2, be independent random variables with values in X and distributions μ i , and let α ij , i, j = 1, 2,…, n, be automorphisms of X. We prove that the independence of n linear forms $ {L_j} = \sum\nolimits_{i = 1}^n {{\alpha_{ij}}} {\xi_i} $ implies that all μ i are shifts of the Haar distributions of a certain subgroup of the group X. This theorem is an analog of the Skitovich–Darmois theorem for finite Abelian groups.

Author(s): Kitzmiller, Kyle; Rathbun, Matt | Abstract: We introduce ideas from geometric group theory related to boundaries of groups. This is a mostly expository paper. We consider the visual boundary of a free abelian group, and show that it is an uncountable set with the trivial topology.

It is known that a (generalized) soluble group whose proper subgroups are abelian is either abelian or finite, and finite minimal non-abelian groups are classified. Here we describe the structure of groups in which every subgroup of infinite index is abelian.

We find classes [Formula: see text] of mixed Abelian groups which satisfy the property that [Formula: see text] if and only if [Formula: see text], whenever F ≤ A is a finite subgroup.

In the representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block AN of the normaliser NG(P) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Strong Version of Brouéʼs Abelian Defect Group Conjecture. In this paper, we prove that the strong version of Brouéʼs abelian defect group conjecture is true for the non-principal 2-block A with an elementary abelian defect group P of order 8 of the sporadic simple Conway group Co3. This result completes the verification of the strong version of Brouéʼs abelian defect group conjecture for all primes p and for all p-blocks of Co3.

Let 𝐺 be a finite group. A subgroup 𝐻 of 𝐺 is said to be S-semipermutable in 𝐺 if 𝐻 permutes with every Sylow 𝑝 -subgroup of 𝐺 with ( 𝑝 , | 𝐻 | ) = 1 . In this paper, we study the influence of S-permutability property of certain abelian subgroups of prime power order of a finite group on its structure.

As an important application of Thompson's theorem [9, Theorem 10.4.2], a finite group is solvable if it has an abelian maximal subgroup. In this article, we mainly investigate the influence of some quantitative properties of abelian subgroups on solvability of finite groups. Some new results are obtained.

The paper deals with the following problem: If a finite abelian 2-group is a direct product of its subsets of cardinality 4, does it follow that at least one of the factors is periodic? Two results are presented. In the first one, the structures of the group and the subsets are restricted but the size of the the group is not. In the second one, the group and the factors are general but the order of the group is 26.

In this paper, for any reduced Abelian group A whose torsion-free rank is infinite, we construct a countable set A(A) of Abelian groups connected with the group A in a definite way and such that for any two different groups B and C from the set A(A) the groups B and C are isomorphic but Hom(B,X) ≅ Hom(C,X) for any Abelian group X. The construction of such a set of Abelian groups is closely connected with Problem 34 from L. Fuchs’ book “Infinite Abelian Groups,” Vol. 1.

We describe the commutator invariant subgroups of a nonreduced abelian group. We find out when all commutator invariant subgroups of a separable group and an algebraically compact torsion-free group are fully invariant and describe the E-centers and E-commutants of these and some other groups.

A subgroup A of a p-group G is said to be soft in G if CG(A) = A and |NG(A/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broue. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the normaliser N G ( P ) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Broue's abelian defect group conjecture. We prove in this paper that Broue's abelian defect group conjecture is true for a non-principal 3-block A with an elementary abelian defect group P of order 9 of the Harada–Norton simple group HN . It then turns out that Broue's abelian defect group conjecture holds for all primes p and for all p-blocks of the Harada–Norton simple group HN .

The shift action on the 2-cocycle group Z2(G,C) of a finite group G with coefficients in a finitely generated abelian group C has several useful applications in combinatorics and digital communications, arising from the invariance of a uniform distribution property of cocycles under the action. In this article, we study the shift orbit structure of the coboundary subgroup B2(G,C) of Z2(G,C). The study is placed within a well-known setting involving the Loewy and socle series of a group algebra over G. We prove new bounds on the dimensions of terms in such series. Asymptotic results on the size of shift orbits are also derived; for example, if C is an elementary abelian p-group, then almost all shift orbits in B2(G,C) are maximal-sized for large enough finite p-groups G of certain classes.