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Journal articles on the topic 'Torsion-free group'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Dorantes-Aldama, Alejandro, and Dmitri Shakhmatov. "Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups." Topology and its Applications 230 (October 2017): 562–77. http://dx.doi.org/10.1016/j.topol.2017.08.020.

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2

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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3

Hossein, Sahleh, and Akbar Alijani Ali. "Extensions of Locally Compact Abelian, Torsion-Free Groups by Compact Torsion Abelian Groups." British Journal of Mathematics & Computer Science 22, no. 4 (2017): 1–5. https://doi.org/10.9734/BJMCS/2017/32966.

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Let <em>X</em> be a compact torsion abelian group. In this paper, we show that an extension of <em>F<sub>p</sub></em> by <em>X</em> splits where <em>F<sub>p</sub></em> is the p-adic number group and p a prime number. Also, we show that an extension of a torsion-free, non-divisible LCA group by <em>X</em> is not split.

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4

Hill, J., P. Hill, and W. Ullery. "Abelian groups that are torsion over their endomorphism rings." Bulletin of the Australian Mathematical Society 64, no. 2 (2001): 255–63. http://dx.doi.org/10.1017/s0004972700039915.

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Using Lambek torsion as the torsion theory, we investigate the question of when an Abelian group G is torsion as a module over its endomorphism ring E. Groups that are torsion modules in this sense are called ℒ-torsion. Among the classes of torsion and truly mixed Abelian groups, we are able to determine completely those groups that are ℒ-torsion. However, the case when G is torsion free is more complicated. Whereas no torsion-free group of finite rank is ℒ-torsion, we show that there are large classes of torsion-free groups of infinite rank that are ℒ-torsion. Nevertheless, meaningful definit

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5

Bastos, Gervasio G., and T. M. Viswanathan. "Torsion-free Abelian Groups, Valuations and Twisted Group Rings." Canadian Mathematical Bulletin 31, no. 2 (1988): 139–46. http://dx.doi.org/10.4153/cmb-1988-021-x.

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AbstractAnderson and Ohm have introduced valuations of monoid rings k[Γ] where k is a field and Γ a cancellative torsion-free commutative monoid. We study the residue class fields in question and solve a problem concerning the pure transcendence of the residue fields.

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6

Dehornoy, Patrick. "The group of fractions of a torsion free lcm monoid is torsion free." Journal of Algebra 281, no. 1 (2004): 303–5. http://dx.doi.org/10.1016/j.jalgebra.2004.03.031.

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7

Amberg, Bernhard, Silvana Franciosi, and Francesco de Giovanni. "Nilpotent-by-Noetherian Factorized Groups." Canadian Mathematical Bulletin 32, no. 4 (1989): 391–403. http://dx.doi.org/10.4153/cmb-1989-057-8.

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AbstractIt is shown that a soluble-by-finite product G = AB of a nilpotent-by-noetherian group A and a noetherian group B is nilpotentby- noetherian. Moreover, a bound for the torsion-free rank of the Fitting factor group of G is given, in terms of the torsion-free rank of the Fitting factor group of A and the torsion-free rank of B.

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8

Parmenter, M. M. "Free torsion-free normal complements in integral group rings." Communications in Algebra 21, no. 10 (1993): 3611–17. http://dx.doi.org/10.1080/00927879308824751.

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9

ALBRECHT, ULRICH, and STEFAN FRIEDENBERG. "A NOTE ON B-COSEPARABLE GROUPS." Journal of Algebra and Its Applications 10, no. 01 (2011): 39–50. http://dx.doi.org/10.1142/s0219498811004410.

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Let B be a torsion-free finitely faithful S-group. This paper investigates the class of all torsion-free Abelian groups A with the property that Ext (A, B) is torsion-free. Our discussion is based on a natural extension of the concept of coseparability.

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10

Faticoni, Theodore G. "Torsion-free Abelian groups torsion over their endomorphism rings." Bulletin of the Australian Mathematical Society 50, no. 2 (1994): 177–95. http://dx.doi.org/10.1017/s0004972700013654.

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We use a variation on a construction due to Corner 1965 to construct (Abelian) groups A that are torsion as modules over the ring End (A) of group endomorphisms of A. Some applications include the failure of the Baer-Kaplansky Theorem for Z[X]. There is a countable reduced torsion-free group A such that IA = A for each maximal ideal I in the countable commutative Noetherian integral domain, End (A). Also, there is a countable integral domain R and a countable. R-module A such that (1) R = End(A), (2) T0 ⊗RA ≠ 0 for each nonzero finitely generated (respectively finitely presented) R-module T0,

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11

Tushev, Anatolii V. "Spectra of conjugated ideals in group algebras of abelian groups of finite rank and control theorems." Glasgow Mathematical Journal 38, no. 3 (1996): 309–20. http://dx.doi.org/10.1017/s0017089500031736.

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Throughout kwill denote a field. If a group Γ acts on aset A we say an element is Γ-orbital if its orbit is finite and write ΔΓ(A) for the subset of such elements. Let I be anideal of a group algebra kA; we denote by I+ the normal subgrou(I+1)∩A of A. A subgroup B of an abelian torsion-free group A is said to be dense in A if A/B is a torsion-group. Let I be an ideal of a commutative ring K; then the spectrum Sp(I) of I is the set of all prime ideals P of K such that I≤P. If R is a ring, M is an R-module and x ɛ M we denote by the annihilator of x in R. We recall that a group Γ is said to have

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12

Rips, Eliyahu, and Yoav Segev. "Torsion-free group without unique product property." Journal of Algebra 108, no. 1 (1987): 116–26. http://dx.doi.org/10.1016/0021-8693(87)90125-6.

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13

Ivanov, S. V., and Roman Mikhailov. "On Zero-divisors in Group Rings of Groups with Torsion." Canadian Mathematical Bulletin 57, no. 2 (2014): 326–34. http://dx.doi.org/10.4153/cmb-2012-036-6.

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AbstractNontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent n ≫ 1 is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.

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14

Tkačenko, Michael G., and Luis M. Villegas-Silva. "Refining connected topological group topologies on Abelian torsion-free groups." Journal of Pure and Applied Algebra 124, no. 1-3 (1998): 281–88. http://dx.doi.org/10.1016/s0022-4049(96)00109-0.

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15

Edjvet, Martin, and James Howie. "On singular equations over torsion-free groups." International Journal of Algebra and Computation 31, no. 03 (2021): 551–80. http://dx.doi.org/10.1142/s0218196721500272.

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We prove a Freiheitssatz for one-relator products of torsion-free groups, where the relator has syllable length at most [Formula: see text]. This result has applications to equations over torsion-free groups: in particular a singular equation of syllable length at most [Formula: see text] over a torsion-free group has a solution in some overgroup.

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16

GROVES, DANIEL P. "FREE GROUPS OF OUTER COMMUTATOR VARIETIES OF GROUPS." Journal of the London Mathematical Society 64, no. 2 (2001): 423–35. http://dx.doi.org/10.1112/s002461070100237x.

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If F is a free group, 1 < i [les ] j [les ] 2i and i [les ] k [les ] i + j + 1 then F/[γj(F), γi(F), γk(F)] is residually nilpotent and torsion-free. This result is extended to 1 < i [les ] j [les ] 2i and i [les ] k [les ] 2i + 2j. It is proved that the analogous Lie rings, L/[Lj, Li, Lk] where L is a free Lie ring, are torsion-free. Candidates are found for torsion in L/[Lj, Li, Lk] whenever k is the least of {i, j, k}, and the existence of torsion in L/[Lj, Li, Lk] is proved when i, j, k [les ] 5 and k is the least of {i, j, k}.

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17

BREAZ, SIMION. "Finite torsion-free rank endomorphism rings." Carpathian Journal of Mathematics 31, no. 1 (2015): 39–43. http://dx.doi.org/10.37193/cjm.2015.01.04.

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18

Tausk, Daniel V. "A Locally Compact Non Divisible Abelian Group Whose Character Group Is Torsion Free and Divisible." Canadian Mathematical Bulletin 56, no. 1 (2013): 213–17. http://dx.doi.org/10.4153/cmb-2011-146-4.

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AbstractIt was claimed by Halmos in 1944 that if G is a Hausdorff locally compact topological abelian group and if the character group of G is torsion free, then G is divisible. We prove that such a claim is false by presenting a family of counterexamples. While other counterexamples are known, we also present a family of stronger counterexamples, showing that even if one assumes that the character group of G is both torsion free and divisible, it does not follow that G is divisible.

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19

ANDRUSZKIEWICZ, R. R., and M. WORONOWICZ. "A TORSION-FREE ABELIAN GROUP EXISTS WHOSE QUOTIENT GROUP MODULO THE SQUARE SUBGROUP IS NOT A NIL-GROUP." Bulletin of the Australian Mathematical Society 94, no. 3 (2016): 449–56. http://dx.doi.org/10.1017/s0004972716000435.

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The first example of a torsion-free abelian group $(A,+,0)$ such that the quotient group of $A$ modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from $2$ to $2^{\aleph _{0}}$ is presented. Ring multiplications on $p$-pure subgroups of the additive group of the r

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20

Chiodo, Maurice, and Rishi Vyas. "Torsion, torsion length and finitely presented groups." Journal of Group Theory 21, no. 5 (2018): 949–71. http://dx.doi.org/10.1515/jgth-2018-0022.

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Abstract We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some {C^{\prime}(1/6)} finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is {C^{\prime}(1/6)} , and thus word-hyperbolic and virtually torsion-free.

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21

Bibi, Mairaj, and Martin Edjvet. "Solving equations of length seven over torsion-free groups." Journal of Group Theory 21, no. 1 (2018): 147–64. http://dx.doi.org/10.1515/jgth-2017-0032.

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AbstractPrishchepov [16] proved that all equations of length at most six over torsion-free groups are solvable. A different proof was given by Ivanov and Klyachko in [12]. This supports the conjecture stated by Levin [15] that any equation over a torsion-free group is solvable. Here it is shown that all equations of length seven over torsion-free groups are solvable.

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22

GÖBEL, R., K. KAARLI, L. MÁRKI, and S. L. WALLUTIS. "ENDOPRIMAL TORSION-FREE SEPARABLE ABELIAN GROUPS." Journal of Algebra and Its Applications 03, no. 01 (2004): 61–73. http://dx.doi.org/10.1142/s0219498804000708.

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We give a characterization for the groups in the title in terms of the graph structure of the critical types occurring in the group. Moreover, we give an example of arbitrarily large endoprimal indecomposable groups.

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23

SIDKI, SAID. "THE BINARY ADDING MACHINE AND SOLVABLE GROUPS." International Journal of Algebra and Computation 13, no. 01 (2003): 95–110. http://dx.doi.org/10.1142/s0218196703001328.

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We prove that any solvable subgroup K of automorphisms of the binary tree, which contains the binary adding machine is an extension of a torsion-free metabelian group by a finite 2-group. If the group K is moreover nilpotent then it is torsion-free abelian.

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24

Solie, Brent B. "Primitivity of group rings of non-elementary torsion-free hyperbolic groups." Journal of Algebra 493 (January 2018): 438–43. http://dx.doi.org/10.1016/j.jalgebra.2017.09.034.

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25

Smith, Howard. "On torsion-free hypercentral groups with all subgroups subnormal." Glasgow Mathematical Journal 31, no. 2 (1989): 193–94. http://dx.doi.org/10.1017/s0017089500007734.

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There is no example known of a non-nilpotent, torsion-free group which has all of its subgroups subnormal. It was proved in [3] that a torsion-free solvable group with all of its proper subgroups subnormal and nilpotent is itself nilpotent, but that seems to be the only published result in this area which is concerned specifically with torsion-free groups. Possibly the extra hypothesis that the group be hypercentral is sufficient to ensure nilpotency, though this is certainly not the case for groups with torsion, as was shown in [7]. The groups exhibited in that paper were seen to have hyperce

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26

Scarparo, Eduardo. "A torsion-free algebraically $\mathrm{C}^*$-unique group." Rocky Mountain Journal of Mathematics 50, no. 5 (2020): 1813–15. http://dx.doi.org/10.1216/rmj.2020.50.1813.

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27

Azam, Saeid, Yoji Yoshii, and Malihe Yousofzadeh. "Jordan tori for a torsion free abelian group." Frontiers of Mathematics in China 10, no. 3 (2014): 477–509. http://dx.doi.org/10.1007/s11464-014-0414-2.

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28

Brodsky, Sergei D., and James Howie. "The universal torsion-free image of a group." Israel Journal of Mathematics 98, no. 1 (1997): 209–28. http://dx.doi.org/10.1007/bf02937335.

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29

Cameron, Peter J., Horacio Guerra, and Šimon Jurina. "The power graph of a torsion-free group." Journal of Algebraic Combinatorics 49, no. 1 (2018): 83–98. http://dx.doi.org/10.1007/s10801-018-0819-1.

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30

Tushev, A. V. "On invariant ideals in group rings of torsion-free minimax nilpotent groups." Researches in Mathematics 31, no. 2 (2023): 56. http://dx.doi.org/10.15421/242315.

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Let $k$ be a field and let $N$ be a nilpotent minimax torsion-free group acted by a solvable group of operators $G$ of finite rank. In the presented paper we study properties of some types of $G$-invariant ideals of the group ring $kN$.

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31

Gildenhuys, D., O. Kharlampovich, and A. Myasnikov. "CSA-groups and separated free constructions." Bulletin of the Australian Mathematical Society 52, no. 1 (1995): 63–84. http://dx.doi.org/10.1017/s0004972700014453.

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A group G is called a CSA-group if all its maximal Abelian subgroups are malnormal; that is, Mx ∩ M = 1 for every maximal Abelian subgroup M and x ∈ G − M. The class of CSA-groups contains all torsion-free hyperbolic groups and all groups freely acting on λ-trees. We describe conditions under which HNN-extensions and amalgamated products of CSA-groups are again CSA. One-relator CSA-groups are characterised as follows: a torsion-free one-relator group is CSA if and only if it does not contain F2 × Z or one of the nonabelian metabelian Baumslag-Solitar groups B1, n = 〈x, y | yxy−1 = xn〉, n ∈ Z ∂

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32

Clifford, A., and R. Z. Goldstein. "Tesselations ofS2and equations over torsion-free groups." Proceedings of the Edinburgh Mathematical Society 38, no. 3 (1995): 485–93. http://dx.doi.org/10.1017/s0013091500019283.

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LetGbe a torsion free group,Fthe free group generated byt. The equationr(t) = 1 is said to have a solution overGif there is a solution in some group that containsG. In this paper we generalize a result due to Klyachko who established the solution when the exponent sum oftis one.

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33

Hillman, Jonathan A. "Deficiencies of lattices in connected Lie groups." Bulletin of the Australian Mathematical Society 65, no. 3 (2002): 393–97. http://dx.doi.org/10.1017/s0004972700020438.

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We complete the determination of the groups of positive deficiency which occur as lattices in connected Lie groups. The torsion free groups among them are 3-mainfold groups. We show that any other torsion free 3-manifold group which is such a lattice is the group of an aspherical closed geometric 3-manifold.

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34

Andruszkiewicz, Ryszard R., and Mateusz Woronowicz. "The Classification of Torsion-free TI-Groups." Algebra Colloquium 29, no. 04 (2022): 595–606. http://dx.doi.org/10.1142/s1005386722000414.

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An abelian group [Formula: see text] is called a [Formula: see text]-group if every associative ring with the additive group [Formula: see text] is filial. The filiality of a ring [Formula: see text] means that the ring [Formula: see text] is associative and all ideals of any ideal of [Formula: see text] are ideals in [Formula: see text]. In this paper, torsion-free [Formula: see text]-groups are described up to the structure of associative nil groups. It is also proved that, for torsion-free abelian groups that are not associative nil, the condition [Formula: see text] implies the indecomposa

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35

Küsmüş, Ömer. "Unit group of integral group ring ℤ(G × C3)". Miskolc Mathematical Notes 25, № 2 (2024): 829. https://doi.org/10.18514/mmn.2024.4666.

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Presenting an explicit descryption of unit group in the integral group ring of a given non-abelian group is a classical and open problem. Let S3 be a symmetric group of order 6 and C3 be a cyclic group of order 3. In this study, we firstly explore the commensurability in unit group of integral group ring ℤ(S3 × C3) by showing the existence of a subgroup as (F55 ⋊ F3) ⋊ (S3∗× C2) where Fρ denotes a free group of rank ρ. Later, we introduce an explicit structure of the unit group in ℤ(S3 × C3) in terms of semi-direct product of torsion-free normal complement of S3 and the group of units in RS3 w

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36

Longobardi, Patrizia, Mercede Maj, Howard Smith, and James Wiegold. "Torsion-free groups isomorphic to all of their non-nilpotent subgroups." Journal of the Australian Mathematical Society 71, no. 3 (2001): 339–48. http://dx.doi.org/10.1017/s1446788700002974.

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AbstractThe main result is that every torsion-free locally nilpotent group that is isomorphic to each of its nonnilpotent subgroups is nilpotent, that is, a torsion-free locally nilpotent group G that is not nilpotent has a non-nilpotent subgroup H that is not isomorphic to G.

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37

Mostovoy, Jacob, José M. Pérez-Izquierdo, and Ivan P. Shestakov. "On torsion-free nilpotent loops." Quarterly Journal of Mathematics 70, no. 3 (2019): 1091–104. http://dx.doi.org/10.1093/qmath/haz010.

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Abstract We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotence of the free loop proved by Higman, it is established, essentially, by the same method.

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GROVES, DANIEL. "SOME PROPERTIES OF FREE GROUPS OF SOME SOLUBLE VARIETIES OF GROUPS." Journal of the London Mathematical Society 63, no. 3 (2001): 592–606. http://dx.doi.org/10.1017/s0024610701002022.

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Let F be a free group, and let γn(F) be the nth term of the lower central series of F. It is proved that F/[γj(F), γi(F), γk(F)] and F/[γj(F), γi(F), γk(F), γl(F)] are torsion free and residually nilpotent for certain values of i, j, k and i, j, k, l, respectively. In the process of proving this, it is proved that the analogous Lie rings are torsion free.

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Andersen, Brooke M., Asher M. Kach, Alexander G. Melnikov, and Reed Solomon. "Jump degrees of torsion-free abelian groups." Journal of Symbolic Logic 77, no. 4 (2012): 1067–100. http://dx.doi.org/10.2178/jsl.7704020.

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40

Bibi, Mairaj, Sajid Ali, Muhammad Shoaib Arif, and Kamaleldin Abodayeh. "Solving singular equations of length eight over torsion-free groups." AIMS Mathematics 8, no. 3 (2023): 6407–31. http://dx.doi.org/10.3934/math.2023324.

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<abstract><p>It was demonstrated by Bibi and Edjvet in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> that any equation with a length of at most seven over torsion-free group can be solvable. This corroborates Levin's <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup> assertion that any equation over a torsion-free group is solvable. It is demonstrated in this article that a singular equation of length eight over torsion-free groups is solvable.</p></abstract>

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41

Bartholdi, Laurent. "On Gardam's and Murray's units in group rings." Algebra and Discrete Mathematics 35, no. 1 (2023): 22–29. http://dx.doi.org/10.12958/adm2053.

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We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices in Gardam's construction that might have appeared arbitrary, and yields more examples of units. We note that all units found up to date exhibit non-trivial symmetry.

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42

Kim, Goansu, and C. Y. Tang. "On the Residual Finiteness of Polygonal Products of Nilpotent Groups." Canadian Mathematical Bulletin 35, no. 3 (1992): 390–99. http://dx.doi.org/10.4153/cmb-1992-052-8.

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AbstractIn general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex gr

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43

GUPTA, C. K., and N. S. ROMANOVSKI. "ON TORSION IN FACTORS OF POLYNILPOTENT SERIES OF A GROUP WITH A SINGLE RELATION." International Journal of Algebra and Computation 14, no. 04 (2004): 513–23. http://dx.doi.org/10.1142/s021819670400189x.

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Let G=F/rF be a group with a single defining relation, r∈Fkm\Fk,m+1, Fij the term of some polynilpotent series of the free group F. We prove: the factors of the corresponding polynilpotent series of the group G are torsion free if and only if r is not a proper power of any element of F modulo Fk,m+1. We also give a description of the lower central series of a group F/[R,R] when F/R is a nilpotent group with torsion free lower central factors.

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44

GUPTA, NARAIN, and SAID SIDKI. "ON TORSION-FREE METABELIAN GROUPS WITH COMMUTATOR QUOTIENTS OF PRIME EXPONENT." International Journal of Algebra and Computation 09, no. 05 (1999): 493–520. http://dx.doi.org/10.1142/s0218196799000308.

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Let G be a torsion-free metabelian group having for commutator quotient, an elementary abelian p-group of rank k. It is shown that k≥3 for all primes p. Examples of such metabelian torsion-free groups are constructed for all primes p and all ranks k≥3, except for p=2, k=3.

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45

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 b

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KHARLAMPOVICH, OLGA, and ALEXEI MYASNIKOV. "DEFINABLE SETS IN A HYPERBOLIC GROUP." International Journal of Algebra and Computation 23, no. 01 (2013): 91–110. http://dx.doi.org/10.1142/s021819671350001x.

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We give a description of definable sets P = (p1,…,pm) in a free non-abelian group F and in a torsion-free non-elementary hyperbolic group G. As a corollary we show that proper non-cyclic subgroups of F and G are not definable. This answers Malcev's question posed in 1965 for F.

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47

Hursey, Robert J. "On nilpotent and polycyclic groups." Bulletin of the Australian Mathematical Society 40, no. 1 (1989): 119–22. http://dx.doi.org/10.1017/s0004972700003567.

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A group G is torsion-free, finitely generated, and nilpotent if and only if G is a supersolvable R-group. An ordered polycylic group G is nilpotent if and only if there exists an order on G with respect to which the number of convex subgroups is one more than the length of G. If the factors of the upper central series of a torsion-free nilpotent group G are locally cyclic, then consecutive terms of the series are jumps, and the terms are absolutely convex subgroups.

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AZHDARI, ZAHEDEH, and MEHRI AKHAVAN-MALAYERI. "ON INNER AUTOMORPHISMS AND CENTRAL AUTOMORPHISMS OF NILPOTENT GROUP OF CLASS 2." Journal of Algebra and Its Applications 10, no. 06 (2011): 1283–90. http://dx.doi.org/10.1142/s0219498811005166.

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Let G be a group and let Aut c(G) be the group of all central automorphisms of G. Let C* = C Aut c(G)(Z(G)) be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper, we prove that if G is a finitely generated nilpotent group of class 2, then C* ≃ Inn (G) if and only if Z(G) is cyclic or Z(G) ≃ Cm × ℤr where [Formula: see text] has exponent dividing m and r is torsion-free rank of Z(G). Also we prove that if G is a finitely generated group which is not torsion-free, then C* = Inn (G) if and only if G is nilpotent group of class 2 and Z(G) is cyclic or Z(G) ≃ Cm × ℤr w

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Albrecht, Ulrich, and Jutta Hausen. "Modules with the quasi-summand intersection property." Bulletin of the Australian Mathematical Society 44, no. 2 (1991): 189–201. http://dx.doi.org/10.1017/s0004972700029610.

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Given a torsion-free abelian group G, a subgroup A of G is said to be a quasi-summand of G if nG ≤ A ⊕ B ≤ G for some subgroup B of G and some positive integer n. If the intersection of any two quasi-summands of G is a quasi-summand, then G is said to have the quasi-summand intersection property. This is a generalisation of the summand intersection property of L. Fuchs. In this note, we give a complete characterisation of the torsion-free abelian groups (in fact, torsion-free modules over torsion-free rings) with the quasi-summand intersection property. It is shown that such a characterisation

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BOGOPOLSKI, O., and E. VENTURA. "ON ENDOMORPHISMS OF TORSION-FREE HYPERBOLIC GROUPS." International Journal of Algebra and Computation 21, no. 08 (2011): 1415–46. http://dx.doi.org/10.1142/s0218196711006601.

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Let H be a torsion-free δ-hyperbolic group with respect to a finite generating set S. From the main result in the paper, Theorem 1.2, we deduce the following two corollaries. First, we show that there exists a computable constant [Formula: see text] such that, for any endomorphism φ of H, if φ(h) is conjugate to h for every element h ∈ H of length up to [Formula: see text], then φ is an inner automorphism. Second, we show a mixed (conjugate/non-conjugate) version of the classical Whitehead problem for tuples is solvable in torsion-free hyperbolic groups.

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