Counting the centralizers of some finite groups

For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is called n-centralizer if #Cent(G) =n, and primitiven-centralizer if # Cent(G)\text = # Cent\text(\fracGZ(G))\text = n\# Cent(G){\text{ = \# }}Cent{\text{(}}\frac{G}{{Z(G)}}){\text{ = }}n. In this paper we compute the number of distinct centralizers of some finite groups and investigate the structure of finite groups with exactly six distinct centralizers. We prove that ifG is a 6-centralizer group then % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGhbaabaGaamOwaiaacIcacaWGhbGaaiykaaaacaqGGaGaeyyrIaKa% aeiiaGqaciaa-readaWgaaWcbaGaa8hoaaqabaGccaGGSaGaaeiiai% aa-feadaWgaaWcbaGaa8hnaaqabaGccaGGSaGaaeiiaiaabQfadaWg% aaWcbaGaaeOmaaqabaGccaqGGaGaey41aqRaaeiiaiaabQfadaWgaa% WcbaGaaeOmaaqabaGccaqGGaGaey41aqRaaeiiaiaabQfadaWgaaWc% baGaaeOmaaqabaGccaqGGaGaae4BaiaabkhacaqGGaGaaeOwamaaBa% aaleaacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaa% leaacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaale% aacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaaleaa% caqGYaaabeaaaaa!62C4!\[\frac{G}{{Z(G)}}{\text{ }} \cong {\text{ }}D_8 ,{\text{ }}A_4 ,{\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ or Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} \] .     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

Partitions and centralizers in the theory of finite groups

This paper is the author's abstract of a dissertation for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended on February 10, 1969 at a united council for adjudicating academic degrees in mathematical sciences at the A. M. Gor'kii Urals State University. Official opponents: Corresponding Member of the Academy of Sciences of the USSR M. I. Kargapolov, Corresponding Member of the Academy of Sciences of the Ukrainian SSR S. N. Chernikov, Doctor of Physicomathematical Sciences Professor B. I. Plotkin, and Doctor of Physicomathematical Sciences V. S. Charin.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 499–511, October, 1969.     

A solubility criterion for finite groups

It is proved that a finite group is soluble if the normalizer of every cyclic subgroup of prime-power order has a soluble supplement.Received: 31 January 2005     

On finite groups with exactly seven element centralizers

For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is calledn-centralizer if #Cent(G) =n, and primitive n-centralizer if $\# Cent(G) = \# Cent\left( {\frac{G}{{Z(G)}}} \right) = n$ . The first author in [1], characterized the primitive 6-centralizer finite groups. In this paper we continue this problem and characterize the primitive 7-centralizer finite groups. We prove that a finite groupG is primitive 7-centralizer if and only if $\frac{G}{{Z(G)}} \cong D_{10} $ orR, whereR is the semidirect product of a cyclic group of order 5 by a cyclic group of order 4 acting faithfully. Also, we compute#Cent(G) for some finite groups, using the structure ofG modulu its center.     

Crowns and centralizers of chief factors of finite groups

We give a new proof of Vaserstein’s Pre-stabilization theorem. This theorem describes GL_n (A) ? E(A) when n is just below the stable range for GL_m (A)/E_m (A) The new proof works only for commutative rings (or ideals in such rings) but it does not need assumptions on Krull dimension, like the old proofs did. All one needs is the relevant stable range con-dition. The new ideas in the proof come from Vaserstein’s recent treatment of the case n = 2. (See preceding paper).     

On finite groups with a certain number of centralizers

LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is calledn-centralizer if #Cent(G)=n, and primitiven-centralizer if #Cent(G)=#Cent(G/Z(G))=n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and ifG is a finite group such thatG/Z(G)?A₅, then #Cent(G)=22 or 32. Moroever, we prove that A₅ is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A₅ in terms of the number of centralizers     

On finite groups with relatively large centralizers of invariant subgroups

In the paper, the finite groups G are studied for which every invariant subgroup A has the property that |G: AC _G (A)| divides a fixed prime p.     

Criteria for the p-solvability and p-supersolvability of finite groups

Let A, K, and H be subgroups of a group G and K ≤ H. Then we say that A covers the pair (K, H) if AH = AK and avoids the pair (K, H) if A ∩ H = A ∩ K. A pair (K, H) in G is said to be maximal if K is a maximal subgroup of H. In the present paper, we study finite groups in which some subgroups cover or avoid distinguished systems of maximal pairs of these groups. In particular, generalizations of a series of known results on (partial) CAP-subgroups are obtained.