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

Abstract simplicity of complete Kac-Moody groups over finite fields

Let $G$ be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix $A$, as constructed by Tits. It is known that $G$ admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac-Moody group $\hat{G}$ which is defined to be the closure of $G$ in the automorphism group of its building. Our main goal is to determine when complete Kac-Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of $\hat{G}$ was previously known to hold when A is of affine type. We extend this result to many indefinite cases, including all hyperbolic generalized Cartan matrices $A$ of rank at least four. Our proof uses Tits' simplicity theorem for groups with a BN-pair and methods from the theory of pro-$p$ groups.Comment: Final version. The statement and the proof of Theorem 5.2 have been corrected. The main result (Theorem 1.1) now holds under slightly stronger restriction

Sobre grupos radicales localmente finitos con min-p para todo primo p.

SUMARY A group is said to be locally finite if every finite subset of G generates a fi-nite subgroup. The class of locally finite groups is placednear the cross-roads of finite group theory and the general theory of infinite groups. Many theorems about finite groups can be phrased in such a way that their statements still make sense for locally finite groups. However, in general, Sylows Theorems do not hold in the class of locally finite groups and there are a number of generic examples which show that locally finite groups can be very varied and complex. If we restrict our attention to locally finite-soluble groups with min-p for all primes p then the Sylow ¼-subgroups are very well behaved if ¼ or its complementary in the set of all primes is finite. The conjugacy of Sylow p-subgroups in these groups is a very strong condition which have guaranteed the successful development of formation theory and interesting results on Fitting classes in the universe c¯L of all radical locally finite groups with min-p for all primes p. Moreover, using an extension of the Frattini subgroup introduced by Tomkinson, it has been proved a Gasch¨utz-Lubeseder type theorem characterizing saturated formations in this universe. It is therefore appropriate to study the class c¯L of all radical locally finite groups with min-p for all primes p in more detail. In this thesis we have obtained results which help to understand better the groups in this class. Consequently, the unspoken rule is that all groups considered in the three chapters of this thesis belong to the class c¯L. The work is organized as follows. In Chapter 1, we explore the class B of generalized nilpotent groups in the universe c¯L. We obtain that this class behaves in the universe c¯L as the nilpotent groups in the finite universe and we determine the structure of B- groups explicitly. Moreover, we show that the largest normal B-subgroup of a c¯L-group is the Fitting subgroup. This fact allows us to prove some results 1 concerning the Fitting subgroup of a c¯L-group which are extensions of the finite ones. The aim of the last section is to study the injectors associated to the class B. In fact, we obtain a description of the B-injectors similar to the characterization of nilpotent injectors of a finite soluble group. Chapter 2 is devoted to study the local version of the class B. This is a natural generalization of the class of finite p-nilpotent groups. We extend some results of finite groups to the above universe using a local version of a Frattini-like subgroup. In particular, some properties appear relating the Frattini and Fitting subgroups. The injectors associated to this class of generalized p-nilpotent groups are also characterized. Finally, Chapter 3 is concerned with the structure of a radical locally finite group with min-p for all p, G = AB, factorized by two subgroups A and B in the class B. We extend the well-known results of finite products of nilpotent groups to the above universe. We have introduced a Chapter 0 establishing the notation and terminology. It also presents many of the well-known results that will be used throughout this thesis

Commutators and commutator subgroups in profinite groups

Let $G$ be a profinite group. We prove that the commutator subgroup $G'$ is finite-by-procyclic if and only if the set of all commutators of $G$ is contained in a union of countably many procyclic subgroups.Comment: 19 pages, final versio

On groups covered by locally nilpotent subgroups

Let N be the class of pronilpotent groups, or the class of locally nilpotent profinite groups, or the class of strongly locally nilpotent profinite groups. It is proved that a profinite group G is finite-by-N if and only if G is covered by countably many N-subgroups. The commutator subgroup G\ue2\u80\ub2is finite-by-N if and only if the set of all commutators in G is covered by countably many N-subgroups. Here, a group is strongly locally nilpotent if it generates a locally nilpotent variety of groups. According to Zelmanov, a locally nilpotent group is strongly locally nilpotent if and only if it is n-Engel for some positive n

On the residual and profinite closures of commensurated subgroups

The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the residual closure of $H$ in $G$ is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.Comment: 22 page