2018
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On groups with a restriction on normal subgroups
2
2
The structure of infinite groups in which every (proper) normal subgroup is the only one of its cardinality is investigated in the universe of groups without infinite simple sections. The corrisponding problem for finite soluble groups was considered by M. Curzio (1958).
1

1
4


Alessio
Russo
Seconda Universita di Napoli
Seconda Universita di Napoli
Italy
alessio.russo@unina2.it
normal subgroup
soluble group
isomorphism class
[[1] R. Armstrong, Finite groups in which any two subgroups of the same order are isomorphic, Proc Cambridge Philos. Soc., 54 (1958) 18–27. ##[2] M. Curzio, Sugli Ngruppi risolubili, Atti Accad Naz. Lincei Rend. Cl. Sci. Fis. Nat. Mat. Nat., 25 (1958) 447–452. ##[3] F. de Giovanni and A. Russo, A note on groups with few isomorphism classes of subgroups, Colloquium Mathematicum, 144 (2016) 265–271. ##[4] F. de Giovanni and M. Trombetti, Uncountable groups with restrictions on subgroups of large cardinality, J. Algebra, 447 (2016) 383–396. ##[5] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups Springer, Berlin, (1972). ##[6] S. Shelah, On a problem of Kurosh, Jónsson groups and applications, In Word Problem II  the Oxford Book, NorthHolland, Amsterdam, (1972) 373–394.##]
Countably recognizable classes of groups with restricted conjugacy classes
2
2
A group class ${ X}$ is said to be countably recognizable if a group belongs to ${X}$ whenever all its countable subgroups lie in ${X}$. It is proved here that most of the relevant classes of groups defined by restrictions on the conjugacy classes are countably recognizable.
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5
16


Francesco
de Giovanni
Dipartimento di Matematica e Applicazioni  University of Napoli &quot;Federico II&quot;
Dipartimento di Matematica e Applicazioni
Italy
degiovan@unina.it


Marco
Trombetti
Universita di Napoli Federico II,
Universita di Napoli Federico II,
Italy
marco.trombetti@unina.it
conjugacy class
countable recognizability
[[1] R. Baer, Abzählbar erkennbare gruppentheoretische Eigenschaften, Math. Z., 79 (1962) 344–363. ##[2] J. T. Buckley, J. C. Lennox, B. H. Neumann, H. Smith and J. Wiegold, Groups with all subgroups normalbyfinite, J. Austral. Math. Soc. Ser. A, 59 (1995) 384–398. ##[3] F. Catino and F. de Giovanni, Some Topics in the Theory of Groups with Finite Conjugacy Classes, 1, Aracne, Roma, 2015. ##[4] M. R. Dixon, M. J. Evans and H. Smith, Some countably recognizable classes of groups, J. Group Theory, 10 (2007) 641–653. ##[5] S. Franciosi, F. de Giovanni and M. J. Tomkinson, Groups with polycyclicbyfinite conjugacy classes, Boll. Un. Mat. Ital. B (7), 4 (1990) 35–55. ##[6] L. Fuchs, Infinite Abelian Groups, 1, Academic Press, New York, 1970. ##[7] F. de Giovanni, M. Martusciello and C. Rainone, Locally finite groups whose subgroups have finite normal oscillation, Bull. Aust. Math. Soc., 89 (2014) 479–487. ##[8] F. de Giovanni, A. Russo and G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J., 28 (2002) 241–254. ##[9] F. de Giovanni and M. Trombetti, Countable recognizability and nilpotency properties of groups, Rend. Circa. Mat. Palermo, to appear. ##[10] G. Higman, Almost free groups, Proc. London Math. Soc. (3), 1 (1951) 284–290. ##[11] M. I. Kargapolov, Some problems in the theory of nilpotent and solvable groups, Dokl. Akad. Nauk SSSR, 127 (1959) 1164–1166. ##[12] D. H. McLain, Remarks on the upper central series of a group, Proc. Glasgow Math. Assoc., 3 (1956) 38–44. ##[13] B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc., 29 (1954) 236–248. ##[14] B. H. Neumann, Groups with finite classes of conjugate subgroups, Math. Z., 63 (1955) 76–96. ##[15] B. H. Neumann, Group properties of countable character, Selected questions of algebra and logic (collection dedicated to the memory of A. I. Mal’cev), Nauka, Novosibirsk, 1973 197–204. ##[16] R. E. Phillips, fsystems in infinite groups, Arch. Math. (Basel), 20 (1969) 345–355. ##[17] R. E. Phillips, Countably recognizable classes of groups, Rocky Mountain J. Math., 1 (1971) 489–497. ##[18] Y. D. Polovicki˘ı, Groups with extremal classes of conjugate elements, Sibirsk. Mat. Z., 5 (1964) 891–895. ##[19] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer, Berlin, 1972. ##[20] D. J. S. Robinson, On groups with extreme centralizers and normalizers, Adv. Group Theory Appl., 1 (2016) 97–112. ##[21] A. Russo and G. Vincenzi, Groups with many generalized FCsubgroups, Algebra Discrete Math., (2009) 158–166. ##[22] H. Smith, More countably recognizable classes of groups, J. Pure Appl. Algebra, 213 (2009) 1320–1324. ##[23] H. Smith and J. Wiegold, Locally graded groups with all subgroups normalbyfinite, J. Austral. Math. Soc. Ser. A, 60 (1996) 222–227. ##[24] M. J. Tomkinson, FCgroups, 96, Pitman, Boston, 1984.##]
Detecting the prime divisors of the character degrees and the class sizes by a subgroup generated with few elements
2
2
We prove that every finite group $G$ contains a threegenerated subgroup $H$ with the following property: a prime $p$ divides the degree of an irreducible character of $G$ if and only if it divides the degree of an irreducible character of $H.$ There is no analogous result for the prime divisors of the sizes of the conjugacy classes.
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17
21


Andrea
Lucchini
Dipartimento di Matematica
Università di Padova
Dipartimento di Matematica
Università
Italy
lucchini@math.unipd.it
Character degrees
class sizes
[[1] F. Dalla Volta and A. Lucchini, Finite groups that need more generators than any proper quotient, J. Austral. Math. Soc. Ser. A, 64 no. 1 (1998) 82–91. ##[2] F. Dalla Volta, A. Lucchini and F. Morini, On the probability of generating a minimally dgenerated group, J. Aust. Math. Soc., 71 no. 2 (2001) 177–185. ##[3] M. Isaacs, Character theory of finite groups, Pure and Applied Mathematics, no. 69. Academic Press [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1976. ##[4] N. Ito, Some studies on group characters, Nagoya Math. J., 2 (1951) 17–28. ##[5] A. S. Kondrat’ev, On prime graph components of finite simple groups, Math. USSRSb., 67 no. 1 (1990) 235247. ##[6] M. L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math., 38 no. 1 (2008) 175–211. ##[7] A. Lucchini and F. Menegazzo, Generators for finite groups with a unique minimal normal subgroup, Rend. Sem. Mat. Univ. Padova, 98 (1997) 173191. ##[8] A. Lucchini, M. Morigi and P. Shumyatsky, Boundedly generated subgroups of finite groups, Forum Math., 24 no. 4 (2012) 875–887. ##[9] G. O. Michler, Brauers conjectures and the classification of finite simple groups, Lecture Notes in Math, 1178, Springer, Berlin, 1986 129–142. ##[10] J. S. Williams, Prime graph components of finite groups, J. Algebra, 69 (1981) 487–513. ##[11] A. V. Zavarnitsine, On the recognition of finite groups by the prime graph, Algebra and Logic, 45 (2006) 220–231.##]
Conjugacy classes contained in normal subgroups: an overview
2
2
We survey known results concerning how the conjugacy classes contained in a normal subgroup and their sizes exert an influence on the normal structure of a finite group. The approach is mainly presented in the framework of graphs associated to the conjugacy classes, which have been introduced and developed in the past few years. We will see how the properties of these graphs, along with some extensions of the classic Landau's Theorem on conjugacy classes for normal subgroups, have been used in order to classify groups and normal subgroups satisfying certain conjugacy class numerical conditions.
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23
36


Antonio
Beltran
Universitat Jaume I
Universitat Jaume I
Spain
abeltran@mat.uji.es


Maria
Jose Felipe
Universitat Politécnica de València
Universitat Politécnica de Val&egra
Spain
mfelipe@mat.upv.es


Carmen
Melchor
Universitat Jaume I
Universitat Jaume I
Spain
cmelchor@uji.es
Conjugacy classes
normal subgroups
graphs
[[1] Z. Akhlaghi, A. Beltrán, M. J. Felipe and M. Khatami, Structure of normal subgroups with three Gclass sizes, Monatsh. Math., 167 (2012) 1–12. ##[2] Z. Akhlaghi, A. Beltrán, M. J. Felipe and M. Khatami, Normal sections, class sizes and solvability of finite gorups, J. Algebra, 399 (2014) 220–231. ##[3] E. Alemany, A. Beltrán and M. J. Felipe, Nilpotency of normal subgroups having two Gclass sizes, Proc. Amer. Math. Soc., 138 (2011) 2663–2669. ##[4] A. Beltrán and M. J. Felipe, The influence of class sizes on normal subgroups, Proceeding of the “Meeting on Group Theory and its applications, on the occasion of Javiel Otal’s 60 th birthday ”, Biblioteca Rev. Mat. Iberoam., (2011) 45–55. ##[5] A. Beltrán and M. J. Felipe, A generalization on the solvability of finite groups with three class sizes for normal subgroups, London Mathematical Society Lecture Notes, 422 (2013) 173–182. ##[6] A. Beltrán, M. J. Felipe and C. Melchor, Graphs associated to conjugacy classes of normal subgroups in finite groups, J. Algebra, 443 (2015) 335–348. ##[7] A. Beltrán, M. J. Felipe and C. Melchor, Landau’s Theorem on conjugacy classes for normal subgroups, Int. J. Algebra Comput., 26 (2016). ##[8] A. Beltrán, M. J. Felipe and C. Melchor, Normal subgroups whose conjugacy class graph has diameter three, Bull. Aust. Math. Soc., 94 (2016) 266–272. ##[9] A. Beltrán, M. J. Felipe and C. Melchor, Triangles in the graph of conjugacy classes of normal subgroups,##Monatshefte für Mathematik, 182 (2017) 521. ##[10] A. Beltrán, M. J. Felipe and C. Shao, pdivisibility of conjugacy class sizes and normal pcomplements, J. Group Theory, 18 (2015) 133–141. ##[11] H. U. Besche, B. Eick and E. A. O’Brien, A millennium project: constructing small groups, Internat. J. Algebra Comput., 12 (2002) 623644. ##[12] E. A. Bertram, M. Herzog and A. Mann, On a graph related to conjugacy classes of groups, Bull. London Math. Soc., 22 (1990) 569–575. ##[13] A. R. Camina and R. D. Camina, The influence of conjugacy class sizes on the structure of finite groups: a survey, AsianEur. J. Math., 4 (2011) 559–588. ##[14] M. Cartwright, A bound on the number of conjugacy classes of a finite soluble group, J. London Math. Soc., 2 (1987) 229–244. ##[15] M. Cartwright, The number of conjugacy classes of certain finite groups, Quart. J. Math. Oxford Ser., 36 (1985) 393–404. ##[16] C. Casolo and S. Dolfi, The diameter of a conjugacy class graph of finite groups, Bull. London Math. Soc., 28 (1996) 141–148. ##[17] C. Casolo, S. Dolfi and E. Jabara, Finite groups whose noncentral class sizes have the same ppart for some prime p, Isr. J. Math., 192 (2012) 197–219. ##[18] D. Chillag, M. Herzog and A. Mann, On the diameter of a graph related to conjugacy classes of groups, Bull. London Math. Soc., 25 (1993) 255–262. ##[19] S. Dolfi, Arithmetical conditions of the length of the conjugacy classes in finite groups, J. Algebra, 174 (1995) 753–771. ##[20] M. Fang and P. Zhang, Finite groups with graphs containing no triangles, J. Algebra, 264 (2003) 613–619. ##[21] The GAP Group, GAPGroups, Algorithms and Programming, 2015, www.gapsystem.org. ##[22] X. Guo and X. Zhao, On the normal subgroup with exactly two Gconjugacy class sizes, Chin. Ann. Math. B., 30 (2009) 427–432. ##[23] L. S. Kazarin, On groups with isolated conjugacy classes, Izv. Vyssh. Uchebn. Zaved. Mat., 7 (1981) 40–45. ##[24] L. Héthelyi and B. Külshammer. Elements of primepower order and their conjugacy classes in finite groups, J. Aust. Math. Soc., 78 (2005) 291–295. ##[25] E. Landau,Über die Klassenzahl der binären quadratischen Formen von negativer Diskriminante, Math. Ann., 56 (1903) 671–676. ##[26] M. L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math., 38 (2008) 175–211. ##[27] A. Moretó and H. N. Nguyen, Variations of Landau’s theorem for pregular and psingular conjugacy classes, Israel J. Math., 212 (2016) 961–987. ##[28] M. Newman, A bound for the number of conjugacy classes in a group, J. London Math. Soc., 43 (1968) 108–110. ##[29] U. Riese and M. A. Shahabi, Subgroups which are the union of four conjugacy classes, Comm. Algebra., 29 (2001) 695–701. ##[30] M. Shahryari and M. A. Shahabi, Subgroups which are the union of three conjugate classes, J. Algebra., 207 (1998) 326–332.##]
On the relationships between the factors of the upper and lower central series in some nonperiodic groups
2
2
This paper deals with the mutual relationships between the factor group $G/zeta(G)$ (respectively $G/zeta_k(G)$) and $G'$ (respectively $gamma_{k+1}(G)$ and $G^{mathfrak{N}}$). It is proved that if $G/zeta(G)$ (respectively $G/zeta_k(G)$) has finite $0$rank, then $G'$ (respectively $gamma_{k+1}(G)$ and $G^{mathfrak{N}}$) also have finite $0$rank. Furthermore, bounds for the $0$ranks of $G', gamma_{k+1}(G)$ and $G^{mathfrak{N}}$ are obtained.
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37
50


Martyn
Dixon
University of Alabama
University of Alabama
United States of America
mdixon@ua.edu


Leonid
Kurdachenko
National University of Dnepropetrovsk
National University of Dnepropetrovsk
Ukraine
lkurdachenko@yahoo.com.ua


Igor
Subbotin
National University
National University
United States of America
isubboti@nu.edu
finite rank
torsionfree rank
section $p$rank
generalized radical group
[[1] R. Baer, Representations of groups as quotient groups. II. minimal central chains of a group, Trans. Amer. Math. Soc., 58 (1945) 348–389. ##[2] R. Baer, Endlichkeitskriterien für Kommutatorgruppen, Math. Ann., 124 (1952) 161–177. ##[3] R. Baer and H. Heineken, Radical groups of finite abelian subgroup rank, Illinois J. Math., 16 (1972) 533–580. ##[4] A. BallesterBolinches, S. CampMora, L. A. Kurdachenko and J. Otal, Extension of a Schur theorem to groups with a central factor with a bounded section rank, J. Algebra, 393 (2013) 1–15. ##[5] M. R. Dixon, L. A. Kurdachenko and J. Otal, On groups whose factorgroup modulo the hypercentre has finite section prank, J. Algebra, 440 (2015) 489–503. ##[6] M. R. Dixon, L. A. Kurdachenko and N. V. Polyakov, Locally generalized radical groups satisfying certain rank conditions, Ricerche di Matematica, 56 (2007) 43–59. ##[7] M. R. Dixon, L. A. Kurdachenko and A. A. Pypka, The theorems of Schur and Baer: a survey, Int. J. Group##Theory, 4 (2015) 21–32. ##[8] M. De Falco, F. de Giovanni, C. Musella and Y. P. Sysak, On the upper central series of infinite groups, Proc. Amer. Math. Soc., 139 (2011) 385–389. ##[9] V. M. Glushkov, On some questions of the theory of nilpotent and locally nilpotent groups without torsion, Mat. Sbornik N. S., 30 (1952) 79–104. ##[10] L. Kaloujnine,Über gewisse Beziehungen zwischen einer Gruppe und ihren Automorphismen, Bericht über die MathematikerTagung in Berlin, Januar, 1953, Deutscher Verlag der Wissenschaften, Berlin, 1953 164–172. ##[11] M. I. Kargapolov, On solvable groups of finite rank, Algebra i Logika Sem., 1 (1962) 37–44. ##[12] L. A. Kurdachenko and J. Otal, The rank of the factorgroup modulo the hypercenter and the rank of the some hypocenter of a group, Cent. Eur. J. Math., 11 (2013) 1732–1741. ##[13] , Groups with Chernikov factorgroup by hypercentral, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A##Math. RACSAM, 109 (2015) 569–579. ##[14] L. A. Kurdachenko, J. Otal and A. A. Pypka, On some properties of central and generalized series of groups, Reports of the National Academy of Sciences of Ukraine, 1 (2015) 20–24. ##[15] L. A. Kurdachenko, J. Otal and I. Ya. Subbotin, Artinian Modules over Group Rings, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. ##[16] L. A. Kurdachenko and P. Shumyatsky, The ranks of central factor and commutator groups, Math. Proc. Cambridge Philos. Soc., 154 (2013) 63–69. ##[17] A. I. Maltsev, Nilpotent torsionfree groups, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949) 201–212. ##[18] B. H. Neumann, Groups with finite classes of conjugate elements, Proc. London Math. Soc. (3), 1 (1951) 178–187. ##[19] D. J. S. Robinson, A new treatment of soluble groups with a finiteness condition on their abelian subgroups, Bull. London Math. Soc., 8 (1976) 113–129. ##[20] A. Schlette, Artinian, almost abelian groups and their groups of automorphisms, Pacific J. Math., 29 (1969) 403–425. ##[21] I. Schur,Über die Darstellungen der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math., 127 (1904) 20–50. ##[22] D. I. Zaitsev, Hypercyclic extensions of abelian groups, Groups defined by properties of a system of subgroups (Russian), Akad. Nauk Ukrain. SSR, Inst. Mat. Kiev, 152 (1979) 16–37.##]
Regular subgroups, nilpotent algebras and projectively congruent matrices
2
2
In this paper we highlight the connection between certain classes of regular subgroups of the affine group $AGL_n(F)$, $F$ a field, and associative nilpotent $F$algebras of dimension $n$. We also describe how the classification of projective congruence classes of square matrices is equivalent to the classification of regular subgroups of particular shape.
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51
56


Marco
Pellegrini
Universit&agrave; Cattolica del Sacro Cuore
Universit&agrave; Cattolica del Sacro
Italy
marco.a.pellegrini@gmail.com
Regular subgroup
nilpotent algebra
congruent matrices
[1] A. Caranti, F. Dalla Volta and M. Sala, Abelian regular subgroups of the affine group and radical rings, Publ. Math. Debrecen, 69 (2006) 297–308. ##[2] F. Catino, I. Colazzo and P. Stefanelli, On regular subgroups of the affine group, Bull. Aust. Math. Soc., 91 (2015) 76–85. ##[3] F. Catino, I. Colazzo and P. Stefanelli, Regular subgroups of the affine group and asymmetric product of radical braces, J. Algebra, 455 (2016) 164–182. ##[4] P. Heged˝ us, Regular subgroups of the affine group, J. Algebra, 225 (2000) 740–742. ##[5] I. M. Isaacs, Characters of groups associated with finite algebras, J. Algebra, 177 (1995) 708–730. ##[6] M. A. Pellegrini, Isomorphism classes of four dimensional nilpotent associative algebras over a field,##https://arxiv.org/abs/1702.00143. ##[7] M. A. Pellegrini and M. C. Tamburini Bellani, More on regular subgroups of the affine group, Linear Algebra Appl., 505 (2016) 126–151. ##[8] M. A. Pellegrini and M. C. Tamburini Bellani, Regular subgroups of the affine group with no translations, to appear in J. Algebra, doi:10.1016/j.jalgebra.2017.01.045. ##[9] M. C. Tamburini Bellani, Some remarks on regular subgroups of the affine group, Int. J. Group Theory, 1 (2012) 17–23. ##[10] G. D. Williams, Congruence of (2 × 2) matrices, Discrete Math., 224 (2000) 293–297. ##[11] G. D. Williams, Projective congruence in M3(Fq), Results Math., 41 (2002) 396–402.##]
On groups with two isomorphism classes of central factors
2
2
The structure of groups which have at most two isomorphism classes of central factors ($B_2$groups) are investigated. A complete description of $B_2$groups is obtained in the locally finite case and in the nilpotent case. In addition detailed information is obtained about soluble $B_2$groups. Also structural information about insoluble $B_2$groups is given, in particular when such a group has the derived subgroup satisfying the minimal condition.
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57
64


Serena
Siani
Universitamp;agrave; degli Studi di Salerno
Universitamp;agrave; degli Studi di Salerno
Italy
ssiani@unisa.it
Center
Isomorphism types
locally finite groups
locally graded groups
[[1] F. de Giovanni and D. J. S. Robinson, Groups with finitely many derived subgroups, J. London Math. Soc. (2), 71 (2005) 658–668. ##[2] M. Herzog, P. Longobardi and M. Maj, On the number of commutators in groups, Ischia Group Theory 2004, Amer. Math. Soc., Providence, RI, 402 (2006) 181192. ##[3] J. C. Lennox, H. Smith and J. Wiegold, A problem about normal subgroups, J. Pure Appl. Algebra, 88 (1993) 169–171. ##[4] P. Longobardi, M. Maj, D. J. S. Robinson and H. Smith, On groups with two isomorphism classes of derived##subgroups, Glasgow Math. J., 55 (2013) 655–668. ##[5] P. Longobardi, M. Maj and D. J. S. Robinson, Recent results on groups with few isomorphism classes of derived subgroups, Proc. of ”Group Theory, Combinatorics, and Computing”, Boca RatonFlorida, Contemp. Math., 611 (2014) 121–135. ##[6] P. Longobardi, M. Maj and D. J. S. Robinson, Locally finite groups with finitely many isomorphism classes of derived subgroups, J. Algebra, 393 (2013) 102–119. ##[7] G. A. Miller and H. C. Moreno, Nonabelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc., 4 (1903) 398404. ##[8] D. J. S. Robinson, A course in the theory of groups, SpringerVerlag, 1996. ##[9] H. Smith, On homomorphic images of locally graded groups, Rend. Sem. Mat. Univ. Padova, 91 (1994) 5360. ##[10] H. Smith, J. Wiegold, Groups which are isomorphic to their nonabelian subgroups, Rend. Sem. Mat. Univ. Padova, 97 (1997) 7–16.##]