پیوندهای مفید
اخبار و اعلانات
آمار
| تعداد نشریات | 43 |
| تعداد شمارهها | 1,684 |
| تعداد مقالات | 13,781 |
| تعداد مشاهده مقاله | 32,313,769 |
| تعداد دریافت فایل اصل مقاله | 12,775,994 |
Inertial properties in groups | ||
| International Journal of Group Theory | ||
| مقاله 4، دوره 7، شماره 3، آذر 2018، صفحه 17-62 اصل مقاله (395.95 K) | ||
| نوع مقاله: Ischia Group Theory 2016 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2017.21611 | ||
| نویسندگان | ||
| Ulderico Dardano* 1؛ Dikran Dikranjan2؛ Silvana Rinauro3 | ||
| 1Dipartimento Matematica e Appl., v. Cintia, M.S.Angelo 5a, I-80126 Napoli (Italy) | ||
| 2Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy. | ||
| 3Silvana Rinauro, Dipartimento di Matematica, Informatica ed Economia, Universit`a della Basilicata, Via dell’Ateneo Lucano 10, I-85100 Potenza, Italy. | ||
| چکیده | ||
| Let $G$ be a group and $p$ be an endomorphism of $G$. A subgroup $H$ of $G$ is called $p$-inert if $H^p\cap H$ has finite index in the image $H^p$. The subgroups that are $p$-inert for all inner automorphisms of $G$ are widely known and studied in the literature, under the name inert subgroups. The related notion of inertial endomorphism, namely an endomorphism $p$ such that all subgroups of $G$ are $p$-inert, was introduced in \cite{DR1} and thoroughly studied in \cite{DR2,DR4}. The ``dual" notion of fully inert subgroup, namely a subgroup that is $p$-inert for all endomorphisms of an abelian group $A$, was introduced in \cite{DGSV} and further studied in \cite{Ch+, DSZ,GSZ}. The goal of this paper is to give an overview of up-to-date known results, as well as some new ones, and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra. We survey on classical and recent results on groups whose inner automorphisms are inertial. Moreover, we show how inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces, and can be helpful for the computation of the algebraic entropy of continuous endomorphisms. | ||
| کلیدواژهها | ||
| commensurable؛ inert؛ inertial endomorphism؛ entropy؛ intrinsic entropy؛ scale function؛ growth؛ locally compact group؛ locally linearly compact space؛ Mahler measure؛ Lehmer problem | ||
| مراجع | ||
|
[1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965) 309–319.
[2] R. Baer, Automorphismengruppen von Gruppen mit endlichen Bahnen gleichmssig beschrnkter Mchtigkeit, J. Reine. Angew. Math., 262-263 (1973) 93–119.
[3] V. V. Belyaev, Inert subgroups in infinite simple groups, Sibirsk. Mat. Zh. 34 (1993) 17–23; English translation in: Siberian Math. J., 34 (1993) 606–611.
[4] V. V. Belyaev, Locally finite groups containing a finite inseparable subgroup, Sib. Mat. Zh. 34, (1993) 23-41; English translation in: Siberian Math. J., 34 (1993) 218–232.
[5] V. V. Belyaev, Inert subgroups in simple locally finite groups, Finite and locally finite groups, Istanbul, 1994 213-218, NATO Adv. Sci.Inst. Ser. C Math. Phys. Sci., 471, Kluwer Acad. Publ., Dordrecht, 1995.
[6] V. V. Belyaev, M. Kuzucuoglu and E. Seckin, Totally inert groups, Rend. Sem. Mat. Univ. Padova, 102 (1999) 151–156.
[7] V. V. Belyaev and D. A. Shved, Finitary automorphisms of groups, Proc. Steklov Inst. Math., 267 (2009) S49–S56.
[8] V. V. Belyaev and D. A. Shved, Groups of outer finitary automorphisms, Mat. Zametki, 89 (2011) 635–636; translation in: Math. Notes, 89 (2011) 596–597.
[9] G. M. Bergman, H. W. Lenstra Jr., Subgroups close to normal subgroups, J. Algebra, 127 (1989) 80–97.
[10] F. Berlai, D. Dikranjan and A. Giordano Bruno, Scale function vs Topological entropy, Topology Appl., 160 (2013) 2314–2334.
[11] J. C. Beidleman, H. Heineken, A Note on I-Automorphisms, J. Algebra, 234 (2000) 694–706.
[12] S. Breaz and G. Calugareanu, Strongly inert subgroups of Abelian groups, to appear on Rend. Sem. Mat. Univ. Padova, 1xx (201x).
[13] J. T. Buckley, J. C. Lennox, B. H. Neumann, H. Smith and J. Wiegold, Groups with all subgroups normal-by-finite, J. Austral. Math. Soc. Ser. A, 59 (1995) 384–398.
[14] G. Calugareanu, Strongly invariant subgroups, Glasgow Math. J., 57 (2015) 431–443.
[15] G. Carraro, Invarianti nella categoria dei flussi per spazi vettoriali, Master Thesis, Padua University, 2014.
[16] C. Casolo, Groups with finite conjugacy classes of subnormal subgroups, Rend. Sem. Mat. Univ. Padova, 81 (1989) 107–149.
[17] C. Casolo, Groups in which all subgroups are subnormal-by-finite, Adv. Group Theory Appl., 1 (2016) 33–45.
[18] C. Casolo, U. Dardano and S. Rinauro, Groups in which each subgroup is commensurable with a normal subgroup, submitted, arXiv:1705.02360.
[19] C. Casolo and O. Puglisi, Hirsch-Plotkin radical of stability groups, J. Algebra, 370 (2012) 133–151.
[20] I. Castellano, Topological entropy for linearly compact vector spaces, corank and Bernoulli shifts, work in progress.
[21] I. Castellano and A. Giordano Bruno, Algebraic entropy in locally linearly compact vector spaces, M. Fontana et al. (eds.), Rings, Polynomials, and Modules, Springer International Publishing AG 2017,
[22] I. Castellano and A. Giordano Bruno, Topological entropy on locally linearly compact vector spaces, Topology Appl., to appear.
[23] T. Ceccherini-Silberstein, M. Coornaert and F. Krieger, An analogue of Fekete’s lemma for subadditive functions on cancellative amenable semigroups, J. Anal. Math., 124 (2014) 59–81.
[24] A. R. Chekhlov, Fully inert subgroups of completely decomposable finite rank groups and their commensurability, (in Russian) Vestn. Tomsk. Gos. Univ. Mat. Mekh., (2016) 41 42–50.
[25] A. R. Chekhlov, On Fully Inert Subgroups of Completely Decomposable Groups, Mathematical Notes, 101 (2017) 365–373.
[26] C. D. Cooper, Power automorphisms of a group, Math.Z., 107 (1968) 335–356.
[27] M. Curzio, S. Franciosi and F. de Giovanni, On automorphisms fixing infinite subgroups of groups, Arch. Math. (Basel), 54 (1990) 4–13.
[28] G. Cutolo, Quasi-power automorphisms of infinite groups, Comm. Algebra, 21 (1993) 1009–1022.
[29] G. Cutolo, E. I. Khukhro, J. C. Lennox, S. Rinauro, H. Smith and J. Wiegold, Locally finite groups all of whose subgroups are boundedly finite over their cores, Bull. London Math. Soc., 29 (1997) 563–570.
[30] D. van Dantzig, Studien over topologische Algebra, Dissertation, Amsterdam, 1931.
[31] U. Dardano, On groups with many maximal subgroups, Ricerche Mat., 38 (1989) 261–271.
[32] U. Dardano and C. Franchi, On group automorphisms fixing subnormal subgroups setwise, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 3 (2000) 811–820.
[33] U. Dardano and C. Franchi, A note on groups paralyzing a subgroup series, Rend. Circ. Mat. Palermo (2), 50 (2001) 165–170.
[34] U. Dardano and S. Rinauro, Inertial automorphisms of an abelian group, Rend. Sem. Mat. Univ. Padova, 127 (2012) 213–233.
[35] U. Dardano, S. Rinauro, On the ring of inertial endomorphisms of an abelian group, Ricerche Mat., 63 (2014) S103- S115.
[36] U. Dardano and S. Rinauro, On groups whose subnormal subgroups are inert, Int. J. Group Theory, 4 no. 2 (2015) 17–24.
[37] U. Dardano and S. Rinauro, Inertial endomorphisms of an abelian group, Ann. Mat. Pura Appl. (4), 195 (2016) 219–234.
[38] U. Dardano and S. Rinauro, A group of generalized finitary automorphisms of an abelian group, J. Group Theory, 20 (2017) 347–369.
[39] M. De Falco, F. de Giovanni, C. Musella and N. Trabelsi, Strongly inertial groups, Comm. Algebra, 41 (2013) 2213– 2227.
[40] M. De Falco, F. de Giovanni, C. Musella and Y. P. Sysak, Weakly power automorphisms of groups, Communications in Algebra, (2017).
[41] D. Dikranjan, A. Fornasiero and A. Giordano Bruno, Algebraic entropy for amenable semigroup actions, work in progress.
[42] D. Dikranjan and A. Giordano Bruno, Topological entropy and algebraic entropy for group endomorphisms, in: Proceedings of the International Conference on Topology and Its Applications, (ICTA 2011), Cambridge Scientific Publishers, Cambridge, 2012 133–214.
[43] D. Dikranjan and A. Giordano Bruno, The Pinsker subgroup of an algebraic flow, Jour. Pure Appl. Algebra, 216 (2012) 364–376.
[44] D. Dikranjan and A. Giordano Bruno, The connection between topological and algebraic entropy, Topology Appl., 159 (2012) 2980–2989.
[45] D. Dikranjan and A. Giordano Bruno, Limit free computation of entropy, Rend. Istit. Mat. Univ. Trieste, 44 (2012) 297–312.
[46] D. Dikranjan and A. Giordano Bruno, Entropy in a category, Appl. Categ. Structures, 21 (2013) 67–101
[47] D. Dikranjan and A. Giordano Bruno, Discrete dynamical systems in group theory, Note Mat., 33 (2013) 1–48.
[48] D. Dikranjan and A. Giordano Bruno, The Bridge Theorem for totally disconnected locally compact groups, Topology Appl., 169 (2014) 21–32.
[49] D. Dikranjan and A. Giordano Bruno, Entropy on abelian groups, Advances in Mathematics, 298 (2016) 612–653.
[50] D. Dikranjan, A. Giordano Bruno and L. Salce, Adjoint algebraic entropy, J. Algebra, 324 (2010) 442–463.
[51] D. Dikranjan, A. Giordano Bruno, L. Salce and S. Virili, Fully inert subgroups of divisible Abelian groups, J. Group Theory, 16 (2013) 915–939.
[52] D. Dikranjan, A. Giordano Bruno, L. Salce and S. Virili, Intrinsic algebraic entropy, J. Pure Appl. Algebra, 219 (2015) 2933–2961.
[53] D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy of endomorphisms of abelian groups, Trans. Amer. Math. Soc., 361 (2009) 3401–3434.
[54] D. Dikranjan, L. Salce and P. Zanardo, Fully inert subgroups of free Abelian groups, Period. Math. Hungar., 69 (2014) 69–78.
[55] D. Dikranjan, M. Sanchis and S. Virili, New and old facts about entropy in uniform spaces and topological groups, Topology Appl., 159 (2012) 1916–1942.
[56] M. R. Dixon, M. J. Evans and H. Smith, Groups with all proper subgroups soluble-by-finite rank, J. Algebra 298 (2005) 135–147.
[57] M. Dixon and H. Smith, Embedding groups in locally (soluble-by-finite) simple groups, J. Group Theory, 9 (2006) 383–395.
[58] M. R. Dixon, M. Evans and A. Tortora, On totally inert simple groups, Cent. Eur. J. Math, 8 (2010) 22–25.
[59] S. Franciosi and F. de Giovanni, Groups in which every infinite subnormal subgroup is normal, J. Algebra, 96 (1985) 566–580.
[60] S. Franciosi and F. de Giovanni, Groups whose subnormal nonnormal subgroups have finite index, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 17 (1993) 241–251.
[61] S. Franciosi, F. de Giovanni and M. L. Newell, Groups whose subnormal subgroups are normal-by-finite, Comm. Alg., 23 (1995) 5483–5497.
[62] L. Fuchs, Infinite Abelian Groups, Academic Press, New York - London, 1970–1973.
[63] W. Gasch¨utz, Gruppen, in denen das Normalteilersein transitiv ist, J. Reine Angew. Math., 198 (1957) 87–92.
[64] A. Giordano Bruno and L. Salce, A soft introduction to algebraic entropy, Arabian J. of Math., 1 (2012) 69–87.
[65] A. Giordano Bruno and L. Salce, Adjoint intrinsic algebraic entropy, work in progress.
[66] A. Giordano Bruno and P. Spiga, Some properties of the growth and of the algebraic entropy of group endomorphisms, J. Group Theory, 20 (2017) 763–774.
[67] A. Giordano Bruno and P. Spiga, Milnor-Wolf theorem for the growth of endomorphisms of locally virtually soluble groups, submitted.
[68] A. Giordano Bruno and S. Virili, The Algebraic Yuzvinski Formula, J. Algebra, 423 (2015) 114–147.
[69] A. Giordano Bruno and S. Virili, On the Algebraic Yuzvinski Formula, Topol. Algebra and its Appl., 3 (2015) 86–103.
[70] A. Giordano Bruno and S. Virili, Topological entropy in totally disconnected locally compact groups, Ergodic Theory Dynam. Systems, 37 (2017) 2163–2186.
[71] F. de Giovanni, Some trends in the theory of groups with restricted conjugacy classes, Note Mat., 33 (2013) 71–87.
[72] R. G¨obel and L. Salce, Endomorphism rings with different rank-entropy supports, Q. J. Math., 63 (2012) 381–397.
[73] B. Goldsmith and L. Salce, When the intrinsic algebraic entropy is not really intrinsic, Topol. Algebra Appl., 3 (2015) 45–56.
[74] B. Goldsmith and L. Salce, Algebraic entropies for Abelian groups with applications to the structure of their endomorphism rings: a survey, in Groups, Modules, and Model Theory - Surveys and Recent Developments, Springer 2017 135–175.
[75] B. Goldsmith and L. Salce, ’s realization theorems from the viewpoint of algebraic entropy, Proceedings 2016 Brixen/Graz Conferences, Springer, 2017.
[76] B. Goldsmith, L. Salce and P. Zanardo, Fully inert subgroups of Abelian p-groups, J. Algebra, 419 (2014) 332–349.
[77] B. Goldsmith, L. Salce and P. Zanardo, Fully inert submodules of torsion-free modules over the ring of p-adic integers, Colloq. Math., 136 (2014) 169–178.
[78] J. I. Hall, Finitary linear transformation groups and elements of finite local degree, Arch. Math. (Basel), 50 (1988) 315–318.
[79] P. de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000,
[80] H. Heineken, Groups with neighbourhood conditions for certain lattices, Note Mat., 16 (1996) 131–143.
[81] H. Heineken and L. A. Kurdachenko, Groups with finitely many classes of almost equal normal subgroups, Algebra Colloq., 4 (1997) 329–344.
[82] O. Kegel and D. Schmidt, Existentially closed finitary linear groups, Groups-St. Andrews, 1989, 2 353–362, London Math. Soc. Lecture Note Ser., 160, Cambridge Univ. Press, Cambridge, 1991.
[83] J. C. Lennox and D. J. S. Robinson, The theory of infinite soluble groups, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2004.
[84] U. Meierfrankenfeld, R. E. Phillips and O. Puglisi, Locally solvable finitary linear groups, J. London Math. Soc., 47 (1993) 31–40.
[85] F. Menegazzo and D. J. S. Robinson, A finiteness condition on automorphism groups, Rend. Sem. Mat. Univ. Padova, 78 (1987) 267–277.
[86] W. M¨ohres, Torsionsgruppen, deren Untergruppen alle subnormal sind, Geom. Dedicata, 31 (1989) 237–244.
[87] B. H. Neumann, Groups with finite classes of conjugate subgroups, Math. Z., 63 (1955) 76–96.
[88] D. Palacin and F. O. Wagner, A Fitting theorem for simple theories, Bull. Lond. Math. Soc., 48 (2016) 472–482.
[89] J. Peters, Entropy on discrete Abelian groups, Adv. Math., 33 (1979) 1–13.
[90] R. E. Phillips, The structure of groups of finitary transformations, J. Algebra, 119 (1988) 400–448.
[91] D. J. S. Robinson, Groups in which normality is a transitive relation, Proc. Cambridge Philos. Soc., 60 (1964) 21–38.
[92] D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, 80 Springer-Verlag, New York, 1996.
[93] D. J. S. Robinson, On inert subgroups of a group, Rend. Sem. Mat. Univ. Padova, 115 (2006) 137–159.
[94] L. Salce, P. V´amos and S. Virili, Length functions, multiplicities and algebraic entropy, Forum Math., 25 (2013) 255–282.
[95] L. Salce and S. Virili, Two new proofs concerning the intrinsic algebraic entropy, Comm. Algebra, to appear.
[96] L. Salce and P. Zanardo, A general notion of algebraic entropy and the rank entropy, Forum Math., 21 (2009) 579–599.
[97] G. Schlichting, Operationen mit periodischen Stabilisatoren. Arch. Math. (Basel), 34 (1980) 97–99.
[98] D. Shved, On the structure of groups of virtually trivial automorphisms, Comm. Algebra, 45 (2017) 184–1852.
[99] H. Smith and J. Wiegold, Locally graded groups with all subgroups normal-by- finite, J. Austral. Math. Soc. Ser. A , 60 (1996) 222–227.
[100] W. Specht and H. Heineken, Gruppen mit endlicher Komponentenzahl fastgleicher Untergruppen, Math. Nachr., 134 73–82.
[101] I. Ya. Subbotin, On the ZD-coradical of a KI-group, (Russian) Vychisl. Prikl. Mat. (Kiev), 75 (1991) 120–124; translation in J. Math. Sci., 72 (1994) 3149–3151.
[102] S. Virili, Entropy for endomorphisms of LCA groups, Topology Appl., 159 (2012) 2546–2556.
[103] S. Virili, Algebraic and topological entropy of group actions, preprint.
[104] B. A. F. Wehrfritz, Finite-finitary groups of automorphisms, J. Algebra Appl., 1 (2002) 375–389.
[105] B. A. F. Wehrfritz, On generalized finitary groups, J. Algebra, 247 (2002) 707–727.
[106] M. D. Weiss, Algebraic and other entropies of group endomorphisms, Math. Systems Theory, 8 (1974/75) 243–248.
[107] G. A. Willis, The structure of totally disconnected locally compact groups, Math. Ann., 300 (1994) 341–363.
[108] G. A. Willis, Further properties of the scale function on a totally disconnected group, J. Algebra, 237 (2001) 142–164.
[109] S. Yuzvinski, Metric properties of endomorphisms of compact groups, English Translation: Amer. Math. Soc. Transl. (2), 66 (1968) 63–98.
[110] G. Zacher, Una caratterizzazione reticolare della finitezza dell’indice di un sottogruppo in un gruppo, Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 69 (1980) 317–323.
[111] A. E. Zalesskii, Groups of bounded automorphisms of groups (in Russian), Dokl. Akad. Nauk BSSR 19 (1975) 681–684. | ||
|
آمار تعداد مشاهده مقاله: 672 تعداد دریافت فایل اصل مقاله: 615 |
||