Locally finite groups and their subgroups with small centralizers (original) (raw)

CENTRALIZERS OF p-SUBGROUPS IN SIMPLE LOCALLY FINITE GROUPS

Glasgow Mathematical Journal, 2019

In Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group with an elementary abelian p-subgroup A of order strictly greater than p2 such that CG(A) is Chernikov and for every non-identity α ∈ A the centralizer CG(α) does not involve an infinite simple group, then G is almost locally soluble. This result is a consequence of another result proved in Ersoy et al. [J. Algebra481 (2017), 1–11], namely: if G is a simple locally finite group with an elementary abelian group A of automorphisms acting on it such that the order of A is greater than p2, the centralizer CG(A) is Chernikov and for every non-identity α ∈ A the set of fixed points CG(α) does not involve an infinite simple groups then G is finite. In this paper, we improve this result about simple locally finite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer CG(P) is Chernikov an...

P-soluble linear groups with Sylow p-intersections of small rank

Journal of Algebra, 1979

/O,(G) is not metacyclic, then ,(G)j < pw where w is the bound obtained by Winter for the case P cyciic und of the same order of magnitude in n cannot hold for f/O,(G) meta-. The proof of this result is based on an analysis of the possible structures of P-the only result specifically on the structure of p-soluble linear groups used is Winter's Theorem ([IO]). A theorem of Blackburn ([4] 111 (12.4)) classifying p-groups (p > 3) with no elementary abelian normal subgroups of order p3 is fundamental to the proof. We feel that the methods developed in the proof may be useful in dealing with related questions on p-soluble linear groups. The notation is standard (cf. [2], [4]). I n particular, we recall that O,(G), ,(G) denote the largest normal p-subgroup, p'-subgroup, respectively, of G. i G I13 denotes thep-share of 1 G j. Th e rank k of a finite p-group P is defined In this paper by ph is the maximum of the orders of elementary abelian subgroups of P, We define for prime p and natural number n, m,(n) by u,(n) = 4813 if p=2 = BPi(P-1) if p > 2 is a Fermat prime ==n if p > 2 is not a Fermat prime. Finally, C denotes the complex number field. I

Centralizers of subgroups in simple locally finite groups

Journal of Group Theory, 2012

Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite K-semisimple subgroups. Namely let G be a non-linear simple locally finite group which has a Kegel sequence K ¼ fðG i ; 1Þ : i A Ng consisting of finite simple subgroups. Then for any finite subgroup F consisting of K-semisimple elements in G, the centralizer C G ðF Þ has an infinite abelian subgroup A isomorphic to a direct product of Z pi for infinitely many distinct primes p i. Moreover we prove that if G is a non-linear simple locally finite group which has a Kegel sequence K ¼ fðG i ; 1Þ : i A Ng consisting of finite simple subgroups G i and F is a finite K-semisimple subgroup of G, then C G ðF Þ involves an infinite simple non-linear locally finite group provided that the finite fields k i over which the simple group G i is defined are splitting fields for L i , the inverse image of F inĜ i G i for all i A N. The groupĜ i G i is the inverse image of G i in the corresponding universal central extension group.

ON FINITE p-GROUPS WITH ABELIAN AUTOMORPHISM GROUP

International Journal of Algebra and Computation, 2013

We construct, for the first time, various types of specific non-special finite p-groups having abelian automorphism group. More specifically, we construct groups G with abelian automorphism group such that γ 2 (G) < Z(G) < Φ(G), where γ 2 (G), Z(G) and Φ(G) denote the commutator subgroup, the center and the Frattini subgroup of G respectively. For a finite p-group G with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) Z(G) = Φ(G) is elementary abelian; (ii) γ 2 (G) = Φ(G) is elementary abelian, where p is an odd prime. We construct examples to show the existence of groups G with elementary abelian automorphism group for which exactly one of the above two conditions holds true.

On localizations of quasi-simple groups with given countable center

Groups, Geometry, and Dynamics, 2020

A group homomorphism i : H → G is a localization of H, if for every homomorphismn ϕ : H → G there exists a unique endomorphism ψ : G → G such that iψ = ϕ (maps are acting on the right). Göbel and Trlifaj asked in [18, Problem 30.4(4), p. 831] which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e. a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Thévenaz and Viruel. 1 Introduction Simple groups are a good source of many kinds of realization theorems. For example, we know from Droste-Giraudet-Göbel [8] that every group can be expressed as the outer automorphism of some simple group. Using Ol'shanskii [25], Obraztsov [24] proved that every abelian group is the center of some infinite quasi-simple group. In [25] we can find Burnside groups of large prime exponent with many extra properties. In this paper we