# Groups with reality and conjugacy conditions

Document Type : Research Paper

Authors

Dipartimento di Matematica Universit`a di Salerno Via Ponte don Melillo 84084 - Fisciano (SA), Italy

Abstract

Many results were proved on the structure of finite groups with some‎ ‎restrictions on their real elements and on their conjugacy classes‎. ‎We‎ ‎generalize a few of these to some classes of infinite groups‎. ‎We study groups in which real elements are central‎, ‎groups in which real elements are $2$-elements‎, ‎groups in which all non-trivial classes have the same finite size and $FC$-groups with two non-trivial conjugacy class sizes‎.

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#### References

Z. Arad (1974). $pi$-homogeneity and $pi^{prime}$-closure of finite groups. Pacific J. Math.. 51, 1-9 I.S. Ashmanov and A.Yu. Ol'shanskij (1985). On abelian and central extensions of aspherical groups. Sov. Math.. 29, 65-82 A.R. Camina (1974). Finite groups of conjugate rank 2. Nagoya Math. J.. 53, 47-57 A.R. Camina (1972). Arithmetical conditions on the conjugacy class numbers of a finite group. J. London Math. Soc. (2). 5, 127-132 D. Chillag and A. Mann (1998). Nearly odd-order and nearly real finite groups. Comm. Algebra. 26, 2041-2064 D. Chillag and M. Herzog (1990). On the Length of the Conjugacy Classes of finite Groups. J. Algebra. 131, 110-125 M.R. Dixon (1994). Sylow Theory, Formations and Fitting Classes in Locally Finite Groups. World Scientific, Singapore. S. Dolfi and E. Jabara (2009). The structure of finite groups of conjugate rank 2. Bull. London Math. Soc. (5). 41, 916-926 D. Gorenstein (1968). Finite Groups. Harper & Row, New York. P. Hall (1956). Finite-by-nilpotent groups. Proc. Cambridge Philos. Soc.. 52, 611-616 B. Huppert and N. Blackburn (1982). Finite Groups II. Springer, Berlin. K. Ishikawa (2002). On finite p-groups which only two conjugacy lengths. Isr. J. Math.. 129, 119-123 N. Ito (1953). On finite groups with given conjugate types I. Nagoya Math. J.. 6, 17-28 N. Ito (1970). On finite groups with given conjugate types II. Osaka J. Math.. 7, 231-251 S. Iwasaki (1979/80). On finite groups with exactly two real conjugate classes. Arch. Math. (Basel). 33, 512-517 B.H. Neumann (1955). Groups with finite classes of conjugate subgroups. Math. Z.. 63, 76-96 D.J.S. Robinson (1982). A course in Group Theory. Springer, Berlin. M.J. Tomkinson (1984). FC-groups. Pitman, London. L. Verardi (1988). On groups whose noncentral elements have the same finite number of conjugates. Boll. Un. Mat. Ital. A (7). 2, 391-400