Abstract
A group G is called a Cpp-group for a prime number p, if G has elements of order p and the centralizer of every non-trivial p-element of G is a pgroup. In this paper we prove that the only infinite locally finite simple groups that are Cpp-groups are isomorphic either to PSL(2,K) or, if p = 2, to Sz(K), with K a suitable algebraic field over GF(p). Using this fact, we also give some structure theorems for infinite locally finite Cpp-groups.
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S. Astill, C. W. Parker and R. Waldecker, A note on groups in which the centralizer of every element of order 5 is a 5-group, Sibirskiĭ Matematicheskiĭ Zhurnal 53, no. 5 2012, 967–977; translation in Siberian Mathematical Journal 53, no. 5 (2012), 772–780.
N. Blackburn and B. Huppert, Finite Groups. II, Grundlehren der Mathematischen Wissenschaften, Vol. 242, Springer-Verlag, Berlin–New York, 1982.
R. Brandl, Finite groups all of whose elements are of prime power order, Unione Matematica Italiana. Bollettino. A. Serie V 18 1981, 491–493.
G. Chen, A characterization of alternating groups by the set of orders of maximal abelian groups, Sibirskiĭ Matematicheskiĭ Zhurnal 47, no. 3 2006, 718–721; translation in Siberian Mathematical Journal 47, no. 3 2006, 594–596.
A. L. Delgado and Y. Wu, On locally finite groups in which every element has prime power order, Illinois Journal of Mathematics 46 2002, 885–891.
S. Dolfi, E. Jabara and S. Lucido, C55-groups. SibirskiĬ Matematicheskiĭ Zhurnal 45, no. 6 2004, 1285–1298; translation in Siberian Mathematical Journal 45, no. 6 2004, 1053–1062.
L. R. Fletcher, B. Stellmacher and W. B. Stewart, Endliche Gruppen, die kein Element der Ordung 6 enthalten, Quarterly Journal of Mathematics. Oxford 28 (1977), 143–154.
L. M. Gordon, Finite simple groups with no elements of order six, Bulletin of the Australian Mathematical Society 17 1977, 235–246.
K. W. Gruenberg and K. W. Roggenkamp, Decomposition of the augmentation ideal and of the relation modules of a finite group, Proceedings of the London Mathematical Society 31 1975, 149–166.
B. Hartley, Simple locally finite groups in Finite and Locally Finite Groups (Istanbul, 1994), NATO Advances Science Institutes Series C: Mathematical and Physical Sciences, Vol. 471, Kluwer Academic, Dordrecht, 1995, pp. 1–44.
B. Hartley, A general Brauer–Fowler Theorem and centralizers in locally finite groups, Pacific Journal of Mathematics 152 1992, 101–117.
H. Heineken, On groups all of whose elements have prime power order, Mathematical Proceedings of the Royal Irish Academy 106 2006, 191–198.
G. Higman, Groups and rings having automorphisms without non-trivial fixed elements, Journal of the London Mathematical Society 32 1957, 321–334.
G. Higman, Finite groups in which every element has prime power order, Journal of the London Mathematical Society 32 1957, 335–342.
N. Iiyori and H. Yamaki, Prime graph components of the simple groups of Lie type over the field of even characteristic, Journal of Algebra 155 1993, 335–343.
O. H. Kegel and B. A. F. Wehrfritz, Locally Finite Groups, North-Holland Mathematical Library, Vol. 3, North-Holland Publishing Company, Amsterdam–London, 1973.
A. S. Kondratév, Prime graph components of finite simple groups, Matematicheskiĭ Sbornik 180 (6) (1989), 787–797; translation in Mathematics of the USSR-Sbornik 67 1990, 235–247.
M. S. Lucido, Prime graph components of finite almost simple groups, Rendiconti del Seminario Matematico della Universitá di Padova 102 1999, 1–22; Addendum, Rendiconti del SeminarioMatematico della Universitá di Padova 107 1999, 189–190.
P. Martineau, On 2-modular representations of Suzuki groups, American Journal of Mathematics 94 1972, 55–72.
D. S. Passman, Permutation Groups, W. A. Benjamin, New York–Amsterdam, 1968.
D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts inMathematics, Vol.80, Springer-Verlag, Berlin–New York, 1982.
W. B. Stewart, Groups having strongly self-centralizing 3-centralizers, Proceedings of the London Mathematical Society 26 1973, 653–680.
M. Suzuki, Finite groups with nilpotent centralizer, Transactions of the American Mathematical Society 99 1961, 425–470.
M. Suzuki, On a class of doubly transitive groups, Annals of Mathematics 75 (1962) 105–145.
J. S. Williams, Prime graph components of finite groups, Journal of Algebra 69 1981, 487–513.
W. Yang and Z. Zhang, Locally soluble infinite groups in which every element has prime power order, Southeast Asian Bulletin of Mathematics 26 2003, 857–864.
G. Zacher, Sull’ordine di un gruppo finito risolubile somma dei suoi sottogruppi di Sylow, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 20 1956, 171–174.
G. Zacher, Sui gruppi finiti somma dei loro sottogruppi di Sylow, Rendiconti del Seminario Matematico della Universitá di Padova 27 1957, 267–275.
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Costantini, M., Jabara, E. On locally finite Cpp-groups. Isr. J. Math. 212, 123–137 (2016). https://doi.org/10.1007/s11856-015-1276-3
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DOI: https://doi.org/10.1007/s11856-015-1276-3