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1
Buckley, Joseph, and James Wiegold. "Nilpotent Extensions of Abelian p-GROUPS." Canadian Journal of Mathematics 38, no. 5 (1986): 1025–52. http://dx.doi.org/10.4153/cjm-1986-050-9.
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This paper arose out of an attempt to solve the following problem due to Suprunenko [5, Problem 2.77]. For which pairs of abelian groups A, B is every extension of A by B nilpotent? We obtain complete answers when A and B are p-groups and (a) A has finite exponent or (b) B is divisible or (c) A has infinite exponent, is countable and B is non-divisible. The structure of a basic subgroup of A plays a central role in cases (b) and (c).At the outset we must say that the problem is too difficult to solve in complete generality. If G/A ≅ 2?, then the nilpotency of G depends solely on properties of the associated homomorphism θ.B → Aut A. Thus for instance if A is torsion-free and B finite, G is nilpotent if and only if the extension is a central one, and we would need detailed information on finite subgroups of the group Aut A.
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Butkote, R. "Recipotent matrices and abelian groups." Asian-European Journal of Mathematics 08, no. 04 (2015): 1550038. http://dx.doi.org/10.1142/s1793557115500382.
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In [M. Chandramouleeswaran and P. Muralikrishna, The recipotent abelian group, Int. J. Math. Sci. 5(4) (2010) 191–200], the concept of a recipotent matrix was introduced. A matrix [Formula: see text] of order [Formula: see text] is called recipotent matrix if [Formula: see text] where [Formula: see text] is reversible matrix of [Formula: see text], defined by each nonzero element of [Formula: see text] is replaced by its reciprocal. The aim of this paper is to improve and complete that paper.
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Danchev, Peter. "MODULAR ABELIAN GROUP ALGEBRAS." Asian-European Journal of Mathematics 03, no. 02 (2010): 275–93. http://dx.doi.org/10.1142/s1793557110000192.
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Suppose FG is the F-group algebra of an arbitrary multiplicative abelian group G with p-component of torsion Gp over a field F of char (F) = p ≠ 0. Our theorems state thus: The factor-group S(FG)/Gp of all normed p-units in FG modulo Gp is always totally projective, provided G is a coproduct of groups whose p-components are of countable length and F is perfect. Moreover, if G is a p-mixed coproduct of groups with torsion parts of countable length and FH ≅ FG as F-algebras, then there is a totally projective p-group T of length ≤ Ω such that H × T ≅ G × T. These are generalizations to results by Hill-Ullery (1997). As a consequence, if G is a p-splitting coproduct of groups each of which has p-component with length < Ω and FH ≅ FG are F-isomorphic, then H is p-splitting. This is an extension of a result of May (1989). Our applications are the following: Let G be p-mixed algebraically compact or p-mixed splitting with torsion-complete Gp or p-mixed of torsion-free rank one with torsion-complete Gp. Then the F-isomorphism FH ≅ FG for any group H implies H ≅ G. Moreover, letting G be a coproduct of torsion-complete p-groups or G be a coproduct of p-local algebraically compact groups, then [Formula: see text]-isomorphism [Formula: see text] for an arbitrary group H over the simple field [Formula: see text] of p-elements yields H ≅ G. These completely settle in a more general form a question raised by May (1979) for p-torsion groups and also strengthen results due to Beers-Richman-Walker (1983).
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Danchev, Peter V., and Patrick W. Keef. "A note on essential finite indecomposability and thickness in primary abelian groups." Tamkang Journal of Mathematics 44, no. 2 (2013): 163–70. http://dx.doi.org/10.5556/j.tkjm.44.2013.1040.
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We present a new characterization of essentially finitely indecomposable abelian $p$-groups. Parallel ideas are also applied to the socles of groups, especially in the case of groups that are pure-complete. These results are then used to discuss the class of thick abelian $p$-groups.
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Zhang, Qinhai, Xiujuan Sun, Lijian An, and Mingyao Xu. "Finite p-Groups All of Whose Subgroups of Index p2 Are Abelian." Algebra Colloquium 15, no. 01 (2008): 167–80. http://dx.doi.org/10.1142/s1005386708000163.
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MÜLLER, JÜRGEN, and SIDDHARTHA SARKAR. "A STRUCTURED DESCRIPTION OF THE GENUS SPECTRUM OF ABELIAN p-GROUPS." Glasgow Mathematical Journal 61, no. 2 (2018): 381–423. http://dx.doi.org/10.1017/s0017089518000265.
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AbstractThe genus spectrum of a finite group G is the set of all g such that G acts faithfully on a compact Riemann surface of genus g. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the Abelian p-groups. Motivated by earlier work of Talu for odd primes, we develop a general combinatorial method, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of Abelian p-groups, including the reduced minimum genus. In particular, we determine the complete genus spectrum for a large subclass, namely, those having ‘large’ defining invariants. With our method we construct infinitely many counterexamples to a conjecture of Talu, which states that an Abelian p-group is recoverable from its genus spectrum. Finally, we give a series of examples of our method, in the course of which we prove, for example, that almost all elementary Abelian p-groups are uniquely determined by their minimum genus, and that almost all Abelian p-groups of exponent p2 are uniquely determined by their minimum genus and Kulkarni invariant.
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Benabdallah, K., B. Charles, and A. Mader. "Vertical Subgroups of Primary Abelian Groups." Canadian Journal of Mathematics 43, no. 1 (1991): 3–18. http://dx.doi.org/10.4153/cjm-1991-001-0.
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AbstractMotived by an intrinisic necessary condition for the purifiability of subgroups of primary abelian groups due to K. Benabdallah and T. Okuyama we introduce new functors on the category of pairs (G, A),where A is a subgroup of G,to the category of Z/pZ-vector spaces. The vanishing of these functors leads to the notion of vertical subgroup which is a weakening of purity but also an essential component of the latter. In fact, a vertical subgroup is pure if and only if it is neat. We establish various facts about vertical subgroups and “maximal” vertical subgroups and apply the resulting theory to the problem of purifiability. We show that the class of quasi-complete groups is precisely the class of reduced groups in which every subgroup satisfying the intrinsic necessary condition for purifiability is in fact purifiable. This is also the class of reduced p-groups in which the maximal vertical subgroups are precisely the pure subgroups.
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Danchev, Peter. "ISOMORPHISM OF MODULAR GROUP ALGEBRAS OF ABELIAN GROUPS WITH SEMI-COMPLETE p-PRIMARY COMPONENTS." Communications of the Korean Mathematical Society 22, no. 2 (2007): 157–61. http://dx.doi.org/10.4134/ckms.2007.22.2.157.
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Kuzman, Bostjan. "Invariant subspaces of matrix groups and elementary-abelian covers of K4,4." Filomat 25, no. 4 (2011): 37–53. http://dx.doi.org/10.2298/fil1104037k.
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We study lifting conditions for groups of automorphisms of the complete bipartite graph K4,4. In particular, for p?2 a prime we construct, up to isomorphism of projections, all minimal p-elementary abelian covers of K4,4 such that the respective covering projections admit a lift of some arc-transitive subgroup of Aut(K4,4).
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Carlson, Jon F., David J. Hemmer, and Nadia Mazza. "The group of endotrivial modules for the symmetric and alternating groups." Proceedings of the Edinburgh Mathematical Society 53, no. 1 (2010): 83–95. http://dx.doi.org/10.1017/s0013091508000618.
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AbstractWe complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n ≥ p2, the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelian of rank 1 if n ≥ p2 + p and has rank 2 if p2 ≤ n < p2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano.
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Goodrick, John, Byunghan Kim, and Alexei Kolesnikov. "Homology Groups of Types in Model Theory and the Computation of H2(p)." Journal of Symbolic Logic 78, no. 4 (2013): 1086–114. http://dx.doi.org/10.2178/jsl.7804040.
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AbstractWe present definitions of homology groups Hn (p), n ≥ 0, associated to a complete type p. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group H2(p) for strong types in stable theories and show that any profinite abelian group can occur as the group H2 (p).
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Berkovich, Ya G., G. A. Freiman, and Cheryl E. Praeger. "Small squaring and cubing properties for finite groups." Bulletin of the Australian Mathematical Society 44, no. 3 (1991): 429–50. http://dx.doi.org/10.1017/s0004972700029932.
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A group G is said to have the small squaring property on k-sets if |K2| < k2 for all k-element subsets K of G, and is said to have the small cubing property on k-sets if |K3| < k3 for all k-element subsets K. It is shown that a finite nonabelian group with the small squaring property on 3-sets is either a 2-group or is of the form TP with T a normal abelian odd order subgroup and P a nontrivial 2-group such that Q = Cp(T) has index 2 in P and P inverts T. Moreover either P is abelian and Q is elementary abelian, or Q is abelian and each element of P − Q inverts Q. Conversely each group of the form TP as above has the small squaring property on 3-sets. As for the nonabelian 2-groups with the small squaring property on 3-sets, those of exponent greater then 4 are classified and the examples are similar to dihedral or generalised quaternion groups. The remaining classification problem of exponent 4 nonabelian examples is not complete, but these examples are shown to have derived length 2, centre of exponent at most 4, and derived quotient of exponent at most 4. Further it is shown that a nonabelian group G satisfies |K2| < 7 for all 3-element subsets K if and only if G = S3. Also groups with the small cubing property on 2-sets are investigated.
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Veisi, Amir, and Ali Delbaznasab. "Metric spaces related to Abelian groups." Applied General Topology 22, no. 1 (2021): 169. http://dx.doi.org/10.4995/agt.2021.14446.
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<p>When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x, A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ cl<sub>X</sub>A if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e &lt; g ∈ G and d is bounded, then d′ (A, B) &lt; g if and only if A ⊆ N<sub>d</sub>(B, g) and B ⊆ N<sub>d</sub>(A, g), where N<sub>d</sub>(A, g) = {x ∈ X : d(x, A) &lt; g}, d<sub>B</sub>(A) = sup{d(a, B) : a ∈ A} and d′ (A, B) = sup{d<sub>A</sub>(B), d<sub>B</sub>(A)}.</p>
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Danchev, Peter. "Isomorphism Characterization of Divisible Groups in Modular Abelian Group Rings." gmj 16, no. 1 (2009): 49–54. http://dx.doi.org/10.1515/gmj.2009.49.
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Abstract Suppose G is an abelian group with a p-subgroup H and R is a commutative unitary ring of prime characteristic p with trivial nil-radical. We give a complete description up to isomorphism of the maximal divisible subgroups of 1 + I(RG;H) and (1 + I(RG;H))=H, respectively, where I(RG;H) denotes the relative augmentation ideal of the group algebra RG with respect to H. This paper terminates a series of works by the author on the topic, first of which are [Danchev, Rad. Mat. 13: 23–32, 2004] and [Danchev, Bull. Georgian Acad. Sci. 174: 238–242, 2006].
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Thomas, Simon. "On the Complexity of the Classification Problem for Torsion-Free Abelian Groups of Finite Rank." Bulletin of Symbolic Logic 7, no. 3 (2001): 329–44. http://dx.doi.org/10.2307/2687752.
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In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at most n can be naturally identified with the set S (ℚn) of all nontrivial additive subgroups of ℚn. In 1937, Baer [4] solved the classification problem for the class S(ℚ)of rank 1 groups as follows.Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to behx(p) = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ (x) of x is defined to be the function
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Grover, Pooja, and Ashwani K. Bhandari. "EXPLICIT DETERMINATION OF CERTAIN MINIMAL ABELIAN CODES AND THEIR MINIMUM DISTANCES." Asian-European Journal of Mathematics 05, no. 01 (2012): 1250002. http://dx.doi.org/10.1142/s1793557112500027.
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In this paper minimal codes for several classes of non-cyclic abelian groups have been constructed by explicitly determining a complete set of primitive idempotents in the corresponding group algebras. Some classes of non-p-groups have also been considered. The minimum distances of such abelian codes have been discussed and compared to the minimum distances of cyclic codes of same lengths and dimensions over the same field.
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LI, CAI HENG, and LEI WANG. "RELATIVE ELEMENTARY ABELIAN GROUPS AND A CLASS OF EDGE-TRANSITIVE CAYLEY GRAPHS." Journal of the Australian Mathematical Society 100, no. 2 (2015): 241–51. http://dx.doi.org/10.1017/s1446788715000464.
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Motivated by a problem of characterising a family of Cayley graphs, we study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\mathsf{Aut}(G)$. It is shown that such groups correspond to complete multipartite graphs which are normal edge-transitive Cayley graphs.
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FLENNER, JOSEPH, and VINCENT GUINGONA. "CONVEXLY ORDERABLE GROUPS AND VALUED FIELDS." Journal of Symbolic Logic 79, no. 01 (2014): 154–70. http://dx.doi.org/10.1017/jsl.2013.16.
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Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.
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Zhang, Qinhai, and Lifang Wang. "Finite Non-abelian Simple Groups Which Contain a Non-trivial Semipermutable Subgroup." Algebra Colloquium 12, no. 02 (2005): 301–7. http://dx.doi.org/10.1142/s1005386705000295.
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A subgroup H of a group G is called semipermutable if it is permutable with every subgroup K of G with (|H|,|K|)=1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with (p,|H|)=1. In this paper, we classify all finite non-abelian simple groups which contain a non-trivial semipermutable (s-semipermutable) subgroup. As a corollary of our main result, we give a complete answer to an unsolved problem in group theory proposed by V.S. Monakhov in 1990.
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Gluschankof, Daniel. "The elementary class of products of totally ordered abelian group." Journal of Symbolic Logic 56, no. 1 (1991): 295–99. http://dx.doi.org/10.2307/2274920.
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A basic goal in model-theoretic algebra is to obtain the classification of the complete extensions of a given (first-order) algebraic theory.Results of this type, for the theory of totally ordered abelian groups, were obtained first by A. Robinson and E. Zakon [5] in 1960, later extended by Yu. Gurevich [4] in 1964, and further clarified by P. Schmitt in [6].Within this circle of ideas, we give in this paper an axiomatization of the first-order theory of the class of all direct products of totally ordered abelian groups, construed as lattice-ordered groups (l-groups)—see the theorem below. We think of this result as constituing a first step—undoubtedly only a small one—towards the more general goal of classifying the first-order theory of abelian l-groups.We write groups for abelian l-groups construed as structures in the language 〈 ∨, ∧, +, −, 0〉 (“−” is an unary operation). For unproved statements and unexplicated definitions, the reader is referred to [1].
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Barthel, Tobias, J. P. C. Greenlees, and Markus Hausmann. "On the Balmer spectrum for compact Lie groups." Compositio Mathematica 156, no. 1 (2019): 39–76. http://dx.doi.org/10.1112/s0010437x19007656.
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We study the Balmer spectrum of the category of finite $G$-spectra for a compact Lie group $G$, extending the work for finite $G$ by Strickland, Balmer–Sanders, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of $G$. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes $p$.
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Ragnarsson, Kári. "RETRACTIVE TRANSFERS AND p-LOCAL FINITE GROUPS." Proceedings of the Edinburgh Mathematical Society 51, no. 2 (2008): 465–87. http://dx.doi.org/10.1017/s0013091505001719.
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AbstractIn this paper we explore the possibility of defining $p$-local finite groups in terms of transfer properties of their classifying spaces. More precisely, we consider the question, posed by Haynes Miller, of whether an equivalent theory can be obtained by studying triples $(f,t,X)$, where $X$ is a $p$-complete, nilpotent space with a finite fundamental group, $f:BS\to X$ is a map from the classifying space of a finite $p$-group, and $t$ is a stable retraction of $f$ satisfying Frobenius reciprocity at the level of stable homotopy. We refer to $t$ as a retractive transfer of $f$ and to $(f,t,X)$ as a retractive transfer triple over $S$.In the case where $S$ is elementary abelian, we answer this question in the affirmative by showing that a retractive transfer triple $(f,t,X)$ over $S$ does indeed induce a $p$-local finite group over $S$ with $X$ as its classifying space.Using previous results obtained by the author, we show that the converse is true for general finite $p$-groups. That is, for a $p$-local finite group $(S,\mathcal{F},\mathcal{L})$, the natural inclusion $\theta:BS\to X$ has a retractive transfer $t$, making $(\theta,t,|\mathcal{L}|^{\wedge}_p)$ a retractive transfer triple over $S$. This also requires a proof, obtained jointly with Ran Levi, that $|\mathcal{L}|^{\wedge}_p$ is a nilpotent space, which is of independent interest.
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Abdollahi, Alireza, and Mojtaba Jazaeri. "On groups admitting no integral Cayley graphs besides complete multipartite graphs." Applicable Analysis and Discrete Mathematics 7, no. 1 (2013): 119–28. http://dx.doi.org/10.2298/aadm121212027a.
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Let G be a non-trivial finite group, S ? G \ {e} be a set such that if a 2 S, then a-1 ? S and e be the identity element of G. Suppose that Cay(G, S) is the Cayley graph with the vertex set G such that two vertices a and b are adjacent whenever a-1 ? S. An arbitrary graph is called integral whenever all eigenvalues of the adjacency matrix are integers. We say that a group G is Cayley integral simple whenever every connected integral Cayley graph on G is isomorphic to a complete multipartite graph. In this paper we prove that if G is a non-simple group, then G is Cayley integral simple if and only if G ? Zp2 for some prime number p or G ? Z2 x Z2. Moreover, we show that there exist finite non-abelian simple groups which are not Cayley integral simple.
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Point, Francoise. "Finitely generic models of TUH, for certain model companionable theories T." Journal of Symbolic Logic 50, no. 3 (1985): 604–10. http://dx.doi.org/10.2307/2274316.
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The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).We determine the finitely generic elements of ∑UH via the three following conditions on T:(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)
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Diekert, Volker, Olga Kharlampovich, and Atefeh Mohajeri Moghaddam. "SLP compression for solutions of equations with constraints in free and hyperbolic groups." International Journal of Algebra and Computation 25, no. 01n02 (2015): 81–111. http://dx.doi.org/10.1142/s0218196715400056.
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The paper is a part of an ongoing program which aims to show that the problem of satisfiability of a system of equations in a free group (hyperbolic or even toral relatively hyperbolic group) is NP-complete. For that, we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We review some basic results on SLPs and give full proofs in order to keep this fundamental part of the program self-contained. Next we study systems of equations with constraints in free groups and more generally in free products of abelian groups. We show how to compress minimal solutions with extended Parikh-constraints. This type of constraints allows to express semi-linear conditions as e.g. alphabetic information. The result relies on some combinatorial analysis and has not been shown elsewhere. We show similar compression results for Boolean formula of equations over a torsion-free δ-hyperbolic group. The situation is much more delicate than in free groups. As byproduct we improve the estimation of the "capacity" constant used by Rips and Sela in their paper "Canonical representatives and equations in hyperbolic groups" from a double-exponential bound in δ to some single-exponential bound. The final section shows compression results for toral relatively hyperbolic groups using the work of Dahmani: We show that given a system of equations over a fixed toral relatively hyperbolic group, for every solution of length N there is an SLP for another solution such that the size of the SLP is bounded by some polynomial p(s + log N) where s is the size of the system.
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Conidis, Chris J. "Classifying model-theoretic properties." Journal of Symbolic Logic 73, no. 3 (2008): 885–905. http://dx.doi.org/10.2178/jsl/1230396753.
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AbstractIn 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set A ≤T 0′ is nonlow2 if and only if A is prime bounding, i.e., for every complete atomic decidable theory T, there is a prime model computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for sets. Some of these predicates, such as prime bounding, and others involving equivalence structures and abelian p-groups come from model theory, while others involving meeting dense sets in trees and escaping a given function come from pure computability theory.As predicates of A, the original nine properties are equivalent for sets; however, they are not equivalent in general. This article examines the (degree-theoretic) relationship between the nine properties. We show that the nine properties fall into three classes, each of which consists of several equivalent properties. We also investigate the relationship between the three classes, by determining whether or not any of the predicates in one class implies a predicate in another class.
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Cui, Shawn X., Dawei Ding, Xizhi Han, et al. "Kitaev's quantum double model as an error correcting code." Quantum 4 (September 24, 2020): 331. http://dx.doi.org/10.22331/q-2020-09-24-331.
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Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes --- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the ``log dim R'' term is included in the definition of entanglement entropy.
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Leary, I. J. "The mod-p cohomology rings of some p-groups." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 1 (1992): 63–75. http://dx.doi.org/10.1017/s0305004100070766.
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Throughout this paper p denotes an odd prime. The groups we shall consider are central extensions of a cyclic subgroup by Cp ⊕ Cp, and may be presented asThe group P(n) is defined for each n ≥ 3, and has order pn. The group P(3) is the non-abelian group of order p3 and exponent p. The mod-p cohomology rings of the other groups of order p3 have been known for some time (see [14] or [5] for that of the non-abelian metacyclic group), and so this paper completes the calculation of the mod-p cohomology rings of the groups of order p3.
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Goldsmith, Brendan, and Luigi Salce. "Abelian p-groups with minimal full inertia." Periodica Mathematica Hungarica, August 12, 2021. http://dx.doi.org/10.1007/s10998-021-00414-w.
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AbstractThe class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small" endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also studied. A topological condition of the socle and a structural condition of the Jacobson radical of the endomorphism ring of a p-group G, both of which are satisfied by direct sums of cyclic groups and by torsion-complete groups, are shown to be independent of the property of having minimal full inertia. The new examples of fully inert subgroups, which are proved not to be commensurable with fully invariant subgroups, are shown not to be uniformly fully inert.
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Huang, Zhaohong, and Jiangmin Pan. "On Arc-Transitive Metacyclic Covers of Graphs with Order Twice a Prime." Electronic Journal of Combinatorics 25, no. 3 (2018). http://dx.doi.org/10.37236/7146.
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Quite a lot of attention has been paid recently to the characterization and construction of edge- or arc-transitive abelian (mostly cyclic or elementary abelian) covers of symmetric graphs, but there are rare results for nonabelian covers since the voltage graph techniques are generally not easy to be used in this case. In this paper, we will classify certain metacyclic arc-transitive covers of all non-complete symmetric graphs with prime valency and twice a prime order $2p$ (involving the complete bipartite graph ${\sf K}_{p,p}$, the Petersen graph, the Heawood graph, the Hadamard design on $22$ points and an infinite family of prime-valent arc-regular graphs of dihedral groups). A few previous results are extended.
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Swartz, Eric. "Locally 3-Arc-Transitive Regular Covers of Complete Bipartite Graphs." Electronic Journal of Combinatorics 23, no. 2 (2016). http://dx.doi.org/10.37236/3506.
Full textAbstract:
In this paper, locally $3$-arc-transitive regular covers of complete bipartite graphs are studied, and results are obtained that apply to arbitrary covering transformation groups. In particular, methods are obtained for classifying the locally $3$-arc-transitive graphs with a prescribed covering transformation group, and these results are applied to classify the locally $3$-arc-transitive regular covers of complete bipartite graphs with covering transformation group isomorphic to a cyclic group or an elementary abelian group of order $p^2$.
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Arasteh Rad, E., and Urs Hartl. "Uniformizing the Moduli Stacks of Global G-Shtukas." International Mathematics Research Notices, November 18, 2019. http://dx.doi.org/10.1093/imrn/rnz223.
Full textAbstract:
Abstract This is the 2nd in a sequence of articles, in which we explore moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ is a flat affine group scheme of finite type over a smooth projective curve $C$ over a finite field. Global $\mathfrak{G}$-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global $\mathfrak{G}$-shtukas are algebraic Deligne–Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the 1st article we explained the relation between global $\mathfrak{G}$-shtukas and local ${{\mathbb{P}}}$-shtukas, which are the function field analogs of $p$-divisible groups. Here ${{\mathbb{P}}}$ is the base change of $\mathfrak{G}$ to the complete local ring at a point of $C$. When ${{\mathbb{P}}}$ is smooth with connected reductive generic fiber we proved the existence of Rapoport–Zink spaces for local ${{\mathbb{P}}}$-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global $\mathfrak{G}$-shtukas for smooth $\mathfrak{G}$ with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands–Rapoport conjecture for our moduli stacks.
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Ertaş, Nil Orhan, and Rachid Tribak. "Some variations of projectivity." Journal of Algebra and Its Applications, September 4, 2021, 2250236. http://dx.doi.org/10.1142/s021949882250236x.
Full textAbstract:
We prove that a ring [Formula: see text] has a module [Formula: see text] whose domain of projectivity consists of only some injective modules if and only if [Formula: see text] is a right noetherian right [Formula: see text]-ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor [Formula: see text]-modules is closed under direct summands if and only if [Formula: see text] is a right Bass ring. A ring [Formula: see text] is said to have no right max-p-middle class if every right [Formula: see text]-module is either projective or max-poor. It is shown that if a commutative noetherian ring [Formula: see text] has no right max-p-middle class, then [Formula: see text] is the ring direct sum of a semisimple ring [Formula: see text] and a ring [Formula: see text] which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field [Formula: see text] of [Formula: see text] has a proper [Formula: see text]-submodule which is not complete in its [Formula: see text]-topology. Then we show that a commutative noetherian hereditary ring [Formula: see text] has no right max-p-middle class if and only if [Formula: see text] is a semisimple ring.