Centralizers and the maximum size of the pairwise noncommuting elements in finite groups

Yıl 2017, Cilt: 46 Sayı: 2, 193 - 198, 01.04.2017

Seyyed Majid Jafarian Amiri Hojjat Rostami

Öz

In this article, we determine the structure of all nonabelian groups $G$ such that $G$ has the minimum number of the element centralizers among
nonabelian groups of the same order. As an application of this result, we obtain the sharp lower bound for $\omega(G)$ in terms of the order of $G$ where $\omega(G)$ is the maximum size of a set of the pairwise noncommuting elements of $G$.

Anahtar Kelimeler

finite group, centralizer, CA-group

Kaynakça

• A. Abdollahi, S. M. Jafarian. Amiri and A. M. Hassanabadi, Groups with specific number of centralizers, Houston J. Math. 33(1) (2007), 43-57.
• A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algbera 298 (2) (2006), 468-492.
• A. Abdollahi, A. Azad, A. Mohamadi Hasanabadi and M. Zarrin, On the clique numbers of non-commuting graphs of certain groups, Algebra Colloq, 17(4) (2010), 611-620.
• A. R. Ashra, On finite groups with a given number of centralizers, Algebra Colloq. 7(2) (2000), 139-146.
• A. Ballester-Bolinches and J. Cossey, On non-commuting sets in finite soluble CC-groups, Publ. Mat. 56 (2012), 467-471
• S. J. Baishya, On finite groups with specific number of centralizers, International Electronic Journal of Algebra, 13(2013), 53-62.
• S. M. Belcastro and G. J. Sherman, Counting centralizers in finite groups, Math. Mag. 5 (1994), 111-114.
• Y. G. Berkovich and E. M. Zhmu'd, Characters of Finite Groups, Part 1, Transl. Math. Monographs 172, Amer. Math. Soc., Providence. RI, 1998.
• R. Brown, Minimal covers of $S_n$ by abelian subgroups and maximal subsets of pairwise noncommuting elements, J. Combin. Theory Ser. A 49 (1988), 294-307.
• R. Brown, Minimal covers of $S_n$ by abelian subgroups and maximal subsets of pairwise noncommuting elements, II, J. Combin. Theory Ser. A 56 (1991), 285-289.
• E. A. Bertram, Some applications of graph theory to finite groups, Discrete Math. 44(1) (1983), 31-43.
• A. M. Y. Chin, On noncommuting sets in an extraspecial p-group, J. Group Theory 8(2) (2005), 189194.
• A. Y. M. Chin, On non-commuting sets in an extraspecial p-group, J. Group Theory, 8.2 (2005), 189-194.
• S. Dol, M. Herzog and E. Jabara, Finite groups whose noncentral commuting elements have centralizers of equal size, Bull. Aust. Math Soc, 82 (2010), 293-304.
• I. M. Isaacs, Finite group theory, Grad. Stud. Math, vol. 92, Amer. Math. Soc, Providence, RI, 2008.
• The GAP Group, GAP-Groups, Algoritms, and Programming, version 4.4.10, (2007), (http://www.gap-system.org).
• S. M. Jafarian Amiri and H. Madadi, On the maximum number of the pairwise noncom- muting elements in a finite group, J. Algebra Appl, (2016), Vol. 16, No. 1 (2017) 1650197 (9 pages).
• S. M. Jafarian Amiri, H. Madadi and H. Rostami, On 9-centralizer groups, J. Algebra Appl, Vol. 14, No. 1 (2015) 1550003 (13 pages).
• S. M. Jafarian Amiri and H. Rostami, Groups with a few nonabelian centralizers, Publ. Math. Debrecen, 87 (3-4) (2015), 429-437.
• S. M. Jafarian Amiri, H. Madadi and H. Rostami, On F-groups with central factor of order $p^4$, Math. Slovaca, Accepted.
• S. M. Jafarian Amiri, M. Amiri and H. Rostami, Finite groups determined by the number of element centralizers, Comm. Alg., 45(9) (2017), 37923797.
• B. H. Neumann, A problem of Paul Erdos on groups, J. Austral. Math. Soc. Ser. A 21 (1976), 467-472.
• J. Pakianathan, S. Krishnan Shankar, Nilpotent numbers, Amer. Math. Monthly, (2000), 631-634.
• L. Pyber, The number of pairwise noncommuting elements and the index of the centre in a finite group, J. Lond. Math. Soc. 35(2) (1987), 287-295.
• D. J. S. Robinson, A course in the theory of groups, Springer-Verlag New York, 1996.
• R. Schmidt, Zentralisatorverbande endlicher Gruppen, Rend. Sem. Mat. Univ. Padova 44 (1970), 97-131.
• M. Zarrin, Criteria for the solubility of finite groups by its centralizers, Arch. Math. 96 (2011), 225-226.
• M. Zarrin, On element centralizers in finite groups, Arch. Math. 93(2009), 497-503.
• M. Zarrin, On solubility of groups with finitely many centralizers, Bull. Iran. Math. Soc. 39 (2013), 517-521.