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Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups. (English) Zbl 07242787
This paper is concerned with algebraic entropy in locally finite groups. The topic of algebraic entropy has been studied in various papers such as [D. Dikranjan et al., Trans. Am. Math. Soc. 361, No. 7, 3401–3434 (2009; Zbl 1176.20057); D. Dikranjan and A. G. Bruno, Topology Appl. 159, No. 13, 2980–2989 (2012; Zbl 1256.54061)]. In this paper the authors discuss whether the addition theorem for algebraic entropy holds for a group \(G\) in this context. If the additon theorem holds for \(G\), then it is said that \(AT(G)\) holds. In this paper the authors are concerned with whether \(AT(G)\) holds in the case when \(G\) is locally finite. In this case the algebraic entropy of \(G\) can be computed via the algebraic entropies of the finite subgroups. Their main result is that for finitely quasihamiltonian locally finite groups \(G\) it follows that \(AT(G)\) holds. In particular, if \(G\) is locally finite and either an FC-group or quasihamiltonian, then \(AT(G)\) holds.
MSC:
20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth
20E99 Structure and classification of infinite or finite groups
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[1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. · Zbl 0127.13102
[2] I. Castellano and A. Giordano Bruno, Algebraic entropy in locally linearly compact vector spaces, Rings, Polynomials, and Modules, Springer, Cham (2017), 103-127. · Zbl 1391.37005
[3] I. Castellano and A. Giordano Bruno, Topological entropy for locally linearly compact vector spaces, Topology Appl. 252 (2019), 112-144. · Zbl 1422.22009
[4] D. Dikranjan, A. Fornasiero and A. Giordano Bruno, Algebraic entropy for amenable semigroup actions, J. Algebra 556 (2020), 467-546. · Zbl 1453.20074
[5] D. Dikranjan, A. Fornasiero, A. Giordano Bruno and F. Salizzoni, The addition theorem for locally monotileable monoid actions, submitted (2020), https://arxiv.org/abs/2001.02019.
[6] D. Dikranjan and A. Giordano Bruno, The connection between topological and algebraic entropy, Topology Appl. 159 (2012), no. 13, 2980-2989. · Zbl 1256.54061
[7] D. Dikranjan and A. Giordano Bruno, The Pinsker subgroup of an algebraic flow, J. Pure Appl. Algebra 216 (2012), no. 2, 364-376. · Zbl 1247.37014
[8] D. Dikranjan and A. Giordano Bruno, Topological and algebraic entropy on groups, Proceedings Islamabad ICTA 2011, Cambridge Scientific, Cambridge (2012), 133-214. · Zbl 1300.54002
[9] D. Dikranjan and A. Giordano Bruno, Discrete dynamical systems in group theory, Note Mat. 33 (2013), no. 1, 1-48. · Zbl 1280.37023
[10] D. Dikranjan and A. Giordano Bruno, The bridge theorem for totally disconnected LCA groups, Topology Appl. 169 (2014), 21-32. · Zbl 1322.37007
[11] D. Dikranjan and A. Giordano Bruno, Entropy on abelian groups, Adv. Math. 298 (2016), 612-653. · Zbl 1368.37015
[12] D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy for abelian groups, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3401-3434. · Zbl 1176.20057
[13] A. Giordano Bruno, A bridge theorem for the entropy of semigroup actions, Topol. Algebra Appl. 8 (2020), no. 1, 46-57. · Zbl 1439.37017
[14] A. Giordano Bruno, M. Shlossberg and D. Toller, Algebraic entropy on strongly compactly covered groups, Topology Appl. 263 (2019), 117-140. · Zbl 1429.37006
[15] A. Giordano Bruno and P. Spiga, Some properties of the growth and of the algebraic entropy of group endomorphisms, J. Group Theory 20 (2017), no. 4, 763-774. · Zbl 1401.20041
[16] A. Giordano Bruno and P. Spiga, Milnor-Wolf Theorem for group endomorphisms, J. Algebra 546 (2020), 85-118. · Zbl 07148868
[17] A. Giordano Bruno and S. Virili, About the algebraic Yuzvinski formula, Topol. Algebra Appl. 3 (2015), no. 1, 86-103.
[18] A. Giordano Bruno and S. Virili, Algebraic Yuzvinski formula, J. Algebra 423 (2015), 114-147. · Zbl 1351.37066
[19] A. Giordano Bruno and S. Virili, Topological entropy in totally disconnected locally compact groups, Ergodic Theory Dynam. Systems 37 (2017), no. 7, 2163-2186. · Zbl 1380.37032
[20] K. Iwasawa, On the structure of infinite M-groups, Jpn. J. Math. 18 (1943), 709-728. · Zbl 0061.02504
[21] S. A. Juzvinskiĭ, Calculation of the entropy of a group-endomorphism, Sibirsk. Mat. Ž. 8 (1967), 230-239.
[22] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N. S.) 119 (1958), 861-864. · Zbl 0083.10602
[23] D. A. Lind and T. Ward, Automorphisms of solenoids and p-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 411-419. · Zbl 0634.22005
[24] J. Milnor, Problem 5603, Amer. Math. Monthly 75 (1968), 685-686.
[25] F. Napolitani, Sui p-gruppi modulari finiti, Rend. Semin. Mat. Univ. Padova 39 (1967), 296-303. · Zbl 0169.03404
[26] D. G. Northcott and M. Reufel, A generalization of the concept of length, Quart. J. Math. Oxford Ser. (2) 16 (1965), 297-321. · Zbl 0129.02203
[27] O. Ore, Structures and group theory. I, Duke Math. J. 3 (1937), no. 2, 149-174. · JFM 63.0060.01
[28] J. Peters, Entropy on discrete abelian groups, Adv. Math. 33 (1979), no. 1, 1-13. · Zbl 0421.28019
[29] J. Peters, Entropy of automorphisms on L.C.A. groups, Pacific J. Math. 96 (1981), no. 2, 475-488. · Zbl 0478.28010
[30] D. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Grad. Texts in Math. 80, Springer, New York, 1996.
[31] L. Salce, P. Vámos and S. Virili, Length functions, multiplicities and algebraic entropy, Forum Math. 25 (2013), no. 2, 255-282. · Zbl 1286.16002
[32] L. Salce and S. Virili, Two new proofs concerning the intrinsic algebraic entropy, Comm. Algebra 46 (2018), no. 9, 3939-3949. · Zbl 1429.20038
[33] J. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR 124 (1959), 768-771. · Zbl 0086.10102
[34] P. Vámos, Additive functions and duality over Noetherian rings, Quart. J. Math. Oxford Ser. (2) 19 (1968), 43-55. · Zbl 0153.37101
[35] S. Virili, Entropy for endomorphisms of LCA groups, Topology Appl. 159 (2012), no. 9, 2546-2556. · Zbl 1243.22007
[36] S. Virili, Algebraic entropy of amenable group actions, Math. Z. 291 (2019), no. 3-4, 1389-1417. · Zbl 1450.16019
[37] S. Virili, Algebraic and topological entropy of group actions, preprint.
[38] M. D. Weiss, Algebraic and other entropies of group endomorphisms, Math. Systems Theory 8 (1974/75), no. 3, 243-248. · Zbl 0298.28014
[39] W. Xi, M. Shlossberg and D. Toller, Algebraic entropy on topologically quasihamiltonian groups, Topology Appl. 272 (2020), Article ID 107093. · Zbl 1439.54011
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