Suchergebnisse

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3. ; This research is supported by Universitat Jaume I, Grant P11B2012-05, and by the Valencian Government, Proyecto PROMETEOII/2015/011.

1 12 83 4 ; S ; [EN] We study the solvability of a normal subgroup N of a finite group G having exactly three G-conjugacy class sizes. We show that if the set of G-class sizes of N is {1, m, mpa }, with p a prime not dividing m, then N is solvable. Thus, we get a partial extension for normal subgroups on N. Itˆo's theorem on the solvability of groups having exactly three class sizes. This research is supported by the Valencian Government, Proyecto PROMETEO/2011/30, by the Spanish Government, Proyecto MTM2010-19938-C03-02 and the first author is also supported by grant Fundaci´o Bancaixa P11B2010-47. The second author would like to thank the members of the Institute of Mathematics at the University of Aberdeen for their warm hospitality and for many helpful discussions during her visit to this University. Beltrán, A.; Felipe Román, MJ. (2013). Solvability of normal subgroups and G-class sizes. Publicationes Mathematicae Debrecen. 83(4):1-12. http://hdl.handle.net/10251/68264

[EN] We study the solvability of a normal subgroup N of a finite group G having exactly three G-conjugacy class sizes. We show that if the set of G-class sizes of N is {1, m, mpa }, with p a prime not dividing m, then N is solvable. Thus, we get a partial extension for normal subgroups on N. Itˆo's theorem on the solvability of groups having exactly three class sizes. ; This research is supported by the Valencian Government, Proyecto PROMETEO/2011/30, by the Spanish Government, Proyecto MTM2010-19938-C03-02 and the first author is also supported by grant Fundaci´o Bancaixa P11B2010-47. The second author would like to thank the members of the Institute of Mathematics at the University of Aberdeen for their warm hospitality and for many helpful discussions during her visit to this University. ; Beltrán, A.; Felipe Román, MJ. (2013). Solvability of normal subgroups and G-class sizes. Publicationes Mathematicae Debrecen. 83(4):1-12. https://doi.org/10.5486/PMD.2013.5548 ; S ; 1 ; 12 ; 83 ; 4

[EN] If G is a finite group and N is a normal subgroup of G with two C-conjugacy class sizes of elements of prime power order, then we show that N is nilpotent. ; We would like to thank the referee for some simplifications in the original proofs. This research was supported by the Spanish Government, Proyecto MTM2010-19938-C03-02, and by the Valencian Government, Proyecto PROMETEO/2011/30. The first author was also supported by grant Fundacio Caixa-Castello P11B2010-47. ; Beltrán, A.; Felipe Román, MJ. (2012). Normal subgroups and class sizes of elements of prime power order. Proceedings of the American Mathematical Society. 140(12):4105-4109. https://doi.org/10.1090/S0002-9939-2012-11276-6 ; S ; 4105 ; 4109 ; 140 ; 12

1250036 1250043 11 2 ; Senia ; It is shown that if the set of conjugacy class sizes of a finite group G is {1,m,n,mn}, where m, n are positive integers which do not divide each other, then G is up to central factors a {p,q}-group. In particular, G is solvable. This research is supported by the Spanish Government, Proyecto MTM2010-19938-C03-02 and by the Valencian Government, Proyecto PROMETEO/2011/30, and the first author is also supported by grant Fundacio Caixa-Castello P11B2010-47. Beltrán, A.; Felipe Román, MJ. (2012). On the solvability of groups with four class sizes. Journal of Algebra and Its Applications. 11(2):1250036-1250043. doi:10.1142/S0219498811005609

4105 4109 140 12 ; S ; [EN] If G is a finite group and N is a normal subgroup of G with two C-conjugacy class sizes of elements of prime power order, then we show that N is nilpotent. We would like to thank the referee for some simplifications in the original proofs. This research was supported by the Spanish Government, Proyecto MTM2010-19938-C03-02, and by the Valencian Government, Proyecto PROMETEO/2011/30. The first author was also supported by grant Fundacio Caixa-Castello P11B2010-47. Beltrán, A.; Felipe Román, MJ. (2012). Normal subgroups and class sizes of elements of prime power order. Proceedings of the American Mathematical Society. 140(12):4105-4109. https://doi.org/10.1090/S0002-9939-2012-11276-6

It is shown that if the set of conjugacy class sizes of a finite group G is {1,m,n,mn}, where m, n are positive integers which do not divide each other, then G is up to central factors a {p,q}-group. In particular, G is solvable. ; This research is supported by the Spanish Government, Proyecto MTM2010-19938-C03-02 and by the Valencian Government, Proyecto PROMETEO/2011/30, and the first author is also supported by grant Fundacio Caixa-Castello P11B2010-47. ; Beltrán, A.; Felipe Román, MJ. (2012). On the solvability of groups with four class sizes. Journal of Algebra and Its Applications. 11(2):1250036-1250043. https://doi.org/10.1142/S0219498811005609 ; S ; 1250036 ; 1250043 ; 11 ; 2

We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.

[EN] 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. ; Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume I of Castellón, who is financially supported by a predoctoral grant of this university. The first and second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, grant P11B-2015-77. ; Beltrán, A.; Felipe, MJ.; Melchor, C. (2018). Conjugacy classes contained in normal subgroups: an overview. International Journal of Group Theory. 7(1):23-26. https://doi.org/10.22108/IJGT.2017.21216 ; S ; 23 ; 26 ; 7 ; 1

Let N be a normal subgroup of a group G and let p be a prime.We prove that if the p-part of jxGj is a constant for every prime-power order element x 2 N n Z.N /, then N is solvable and has normal p-complement. ; This research is supported by the Valencian Government, Proyecto PROMETEO/2011/30, and by the Spanish Government, Proyecto MTM2010-19938-C03-02. The third author is supported by the research Project NNSF of China (Grant No. 11201401 and No. 11301218) and University of Jinan Research Funds for Doctors (XBS1335 and XBS1336).

LetG be a finite p-solvable group. We prove that ifG has exactly two conjugacy class sizes of p 0 -elements of prime power order, say 1 and m, then m = p aq b , for two distinct primes p and q, and G either has an abelian p-complement or G = PQ A, with P and Q a Sylow p-subgroup and a Sylow q-subgroup of G, respectively, and A is abelian. In particular, we provide a new extension of Itô's theorem on groups having exactly two class sizes for elements of prime power order. ; This research is supported by the Spanish Government, Proyecto MTM2010-19938-C03-02 and the second and third authors are supported by the Valencian Government, Proyecto PROMETEO/2011/30. The second author is also supported by grant Fundacio Caixa-Castell ´ o P11B2010-47.

[EN] Suppose that G is a finite group and K is a non-trivial conjugacy class of G such that KK (-1) = 1 a D a D (-1) with D a conjugacy class of G. We prove that G is not a non-abelian simple group and we give arithmetical conditions on the class sizes determining the solvability and the structure of aOE (c) K > and aOE (c) D >. ; The authors gratefully acknowledge all helpful comments made by the referee. The results in this paper are part of the third author's Ph.D. thesis, and she acknowledges the predoctoral grant PRE-DOC/2015/46, Universitat Jaume I. The first and second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and third authors are also partially supported by Universitat Jaume I, grant P11B2015-77. ; Beltrán, A.; Felipe Román, MJ.; Melchor, C. (2018). Multiplying a conjugacy class by its inverse in a finite group. Israel Journal of Mathematics. 227(2):811-825. https://doi.org/10.1007/s11856-018-1742-9 ; S ; 811 ; 825 ; 227 ; 2 ; E. Adan-Bante, Products of characters with few irreducible constituents, Journal of Algebra 311 (2007), 38–68. ; E. Adan-Bante, Symmetric groups and conjugacy classes, Journal of Group Theory 3 (2008), 371–379. ; Z. Arad and E. Fisman, An analogy between products of two conjugacy classes and products of two irreducible characters in finite groups, Proceedings of the Edinburgh Mathematical Society 30 (1987), 7–22. ; Z. Arad and M. Herzog, Products of Conjugacy Classes in Groups, Lecture Notes in Mathematics, Vol. 1112, Springer-Verlag, Berlin, 1985. ; A. Beltrán, M. J. Felipe and C. Melchor, Squares of real conjugacy classes in finite groups, Annali di Matematica Pura ed Applicata 197 (2018), 317–328. ; R. W. Carter, Finite Groups of Lie Type, Pure and Applied Mathematics (New York), John Wiley & Sons, New York, 1985. ; The GAP Group, GAP - Groups, Algorithms and Programming, Vers. 4.7.7, 2015, https://doi.org/www.gap-system.org ; G. Glauberman, Central elements in core-free groups, Journal of Algebra 4 (1966), 403–420. ...

We survey known results concerning how the conjugacy classes contained in a normal subgroup and their sizes exert an in uence on the normal structure of a nite 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. ; Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume I of Castellón, who is financially supported by a predoctoral grant of this university. The first and second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, grant P11B2015-77.