Orbits of maximal invariant subgroups and solvability of finite groups

Let A and G be finite groups having coprime orders and suppose that A acts on G via automorphisms. We give some solvability criteria for G according to the number of orbits that appear by the action of the fixed point subgroup on the set of maximal A-invariant subgroups of G, and likewise, on the set of non-nilpotent maximal A-invariant subgroups. We also obtain some characterizations and further structure properties of these groups. In the course of our study we prove an independent result concerning maximal factorizations of classical simple groups

New Conditions on Maximal Invariant Subgroups That Imply Solubility

Let G be a finite group and assume that a finite group of automorphisms A acts on G, such that the orders of A and G are relatively prime. We prove that the fact of imposing certain conditions on the set of maximal A-invariant subgroups of G, relating to nilpotency, p-nilpotency, normality or having p’-order, determines properties on the structure of G such as solubility, p-solubility or p-nilpotency

A coprime action version of a solubility criterion of Deskins

Let A and G be finite groups of relatively prime orders and suppose that A acts on G via automorphisms. We demonstrate that if G has a maximal A-invariant subgroup M that is nilpotent and the Sylow 2-subgroup of M has class at most 2, then G is soluble. This result extends, in the context of coprime action, a solubility criterion given by W.E. Deskins

Conditions for Sylow 2-subgroups of the Fixed Point Subgroup Implying Solubility

Let A and G be finite groups and suppose that A acts via automorphisms on G with (|A|, |G|) = 1. We study how certain conditions on the Sylow 2-subgroups of the fixed point subgroup of the action, CG(A), may imply the non-simplicity or solubility of G

p-divisibility of conjugacy class sizes and normal p-complements

The final publication is available at www.degruyter.com. Let N be a normal subgroup of a group G and let p be a prime. We prove that if the p-part of jx Gj 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).Beltrán, A.; Felipe Román, MJ.; Shao, C. (2015). p-divisibility of conjugacy class sizes and normal p-complements. Journal of Group Theory. 18(1):133-141. https://doi.org/10.1515/jgth-2014-003513314118

Class sizes of prime-power order p'-elements and normal subgroups

We prove an extension of the renowned Itô’s theorem on groups having two class sizes in three different directions at the same time: normal subgroups, p′p′-elements and prime-power order elements. Let NN be a normal subgroup of a finite group GG and let pp be a fixed prime. Suppose that |xG|=1|xG|=1 or mm for every qq-element of NN and for every prime q≠pq≠p. Then, NN has nilpotent pp-complements.We are very grateful to the referee, who provided us a significant simplification of the last step of the proof of the main theorem and for many comments which have contributed to improve the paper. C. G. Shao wants to express his deep gratitude for the warm hospitality he has received in the Departamento de Matematicas of the Universidad Jaume I in Castellon, Spain. This research is supported by the Valencian Government, Proyecto PROMETEO/2011/30, by the Spanish Government, Proyecto MTM2010-19938-C03-02. The third author is supported by the research Project NNSF of China (Grant Nos. 11201401 and 11301218) and University of Jinan Research Funds for Doctors (XBS1335 and XBS1336).Beltrán, A.; Felipe Román, MJ.; Shao, C. (2015). Class sizes of prime-power order p'-elements and normal subgroups. Annali di Matematica Pura ed Applicata. 194(5):1527-1533. https://doi.org/10.1007/s10231-014-0432-4S152715331945Akhlaghi, Z., Beltrán, A., Felipe, M.J.: The influence of $$p$$ p -regular class sizes on normal subgroups. J. Group Theory. 16, 585–593 (2013)Alemany, E., Beltrán, A., Felipe, M.J.: Nilpotency of normal subgroups having two $$G$$ G -class sizes. Proc. Am. Math. Soc. 139, 2663–2669 (2011)Alemany, E., Beltrán, A., Felipe, M.J.: Finite groups with two $$p$$ p -regular conjugacy class lengths II. Bull. Aust. Math. Soc. 797, 419–425 (2009)Beltrán, A., Felipe, M.J.: Normal subgroups and class sizes elements of prime-power order. Proc. Am. Math. Soc. 140, 4105–4109 (2012)Beltrán, A.: Action with nilpotent fixed point subgroup. Arch. Math. (Basel) 69, 177–184 (1997)Camina, A.R.: Finite groups of conjugate rank 2. Nagoya Math. J. 53, 47–57 (1974)Casolo, C., Dolfi, S., Jabara, E.: Finite groups whose noncentral class sizes have the same $$p$$ p -part for some prime $$p$$ p . Isr. J. Math. 192, 197–219 (2012)Huppert, B.: Character Theory of Finite groups, vol. 25. De Gruyter Expositions in Mathemathics, Berlin, New York (1998)Kleidman, P., Liebeck, M.: The Subgroup Structure of The Finite Classical Groups. London Mathematical Society Lecture Note Series, 129. Cambridge University Press, Cambridge (1990)Kurzweil, K., Stellmacher, B.: The Theory of Finite Groups. An Introduction. Springer, New York (2004)The GAP Group, GAP—Groups, Algorithms and Programming, Vers. 4.4.12 (2008). http://www.gap-system.orgVasiliev, A.V., Vdovin, E.P.: An adjacency criterion for the prime graph of a finite simple group. Algebra Logic 44(6), 381–406 (2005

Den manliga hegemonin i förändring? En undersökning av manliga och kvinnliga intressen under riksdagsperioderna 1988 till 2010

summary:Let $G$ be a finite group and $\operatorname{nse}(G)$ the set of numbers of elements with the same order in $G$. In this paper, we prove that a finite group $G$ is isomorphic to $M$, where $M$ is one of the Mathieu groups, if and only if the following hold: (1) $|G|=|M|$, (2) $\operatorname{nse}(G)=\operatorname{nse}(M)$

Arithmetical Conditions on Invariant Sylow Numbers

Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. A natural number n is said to be an A-invariant Sylow number if n is the number of A-invariant Sylow p-subgroups of G for some prime p. We investigate how the fact of imposing certain arithmetical conditions on the set of A-invariant Sylow numbers of G may imply the solvability of G

Invariant TI-subgroups and structure of finite groups

Let G be a finite group and assume that a group of automorphisms A is acting on G such that A and G have coprime orders. Recall that a subgroup H of G is said to be a TI-subgroup if it has trivial intersection with its distinct conjugates in G. We study the solubility and other properties of G when we assume that certain invariant subgroups of G are TI-subgroups, precisely when all A-invariant subgroups, all non-nilpotent A-invariant subgroups, and all non-abelian A-invariant subgroups of G, respectively, are TI-subgroups

Restrictions on maximal invariant subgroups implying solvability of finite groups

Suppose that G and A are finite groups such that A acts coprimely on G via automorphisms. It is interesting to investigate the structure and properties of G when we impose some restrictions on its maximal A-invariant subgroups. More precisely, we prove the solvability of G when certain maximal A-invariant subgroups are nilpotent, when all maximal A-invariant subgroups are supersolvable, or when certain arithmetic conditions are imposed on non-nilpotent maximal A-invariant subgroups