共查询到20条相似文献,搜索用时 15 毫秒
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Alexander A. Mahnjov 《代数通讯》2013,41(12):1307-1322
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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 相似文献
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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. 相似文献
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Edward A. Bertram 《Israel Journal of Mathematics》1984,47(4):335-344
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. 相似文献
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Ali Reza Ashrafi 《Journal of Applied Mathematics and Computing》2000,7(1):115-124
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!\[\frac{G}{{Z(G)}}{\text{ }} \cong {\text{ }}D_8 ,{\text{ }}A_4 ,{\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ or Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} \] . 相似文献
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A. I. Starostin 《Mathematical Notes》1969,6(4):754-760
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. 相似文献
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Julio Lafuente 《代数通讯》2013,41(3):657-668
We give a new proof of Vaserstein’s Pre-stabilization theorem. This theorem describes GLn (A) ? E(A) when n is just below the stable range for GLm (A)/Em (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). 相似文献
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Mohammad Zarrin 《Archiv der Mathematik》2011,96(3):225-226
For any group G, let C(G){\mathcal{C}(G)} denote the set of centralizers of G. We say that a group G has n centralizers (G is a Cn{\mathcal{C}_n}-group) if |C(G)| = n{|\mathcal{C}(G)| = n}. In this note, we show that the derived length of a soluble Cn{\mathcal{C}_n}-group (not necessarily finite) is bounded by a function of n. 相似文献
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Let FH be a supersolvable Frobenius group with kernel F and complement H. Suppose that a finite group G admits FH as a group of automorphisms in such a manner that and is nilpotent of class c. We show that G is nilpotent of -bounded class. 相似文献