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Periodic subgroups of projective linear groups in positive characteristic. (English) Zbl 1159.20026

The authors classify the maximal irreducible periodic subgroup of \(\text{PGL}(q,\mathbb{F})\) where \(\mathbb{F}\) is a field of positive characteristic \(p\) transcendental over its prime subfield, \(q\neq p\) is prime, and \(\mathbb{F}^\times\) has an element of order \(q\). This classification problem is more tractable than a similar problem in characteristic zero.

MSC:

20H20 Other matrix groups over fields
20E07 Subgroup theorems; subgroup growth
20E28 Maximal subgroups

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References:

[1] Bácskai Z., Finite irreducible monomial groups of small prime degree, Ph.D. thesis, Australian National University, 1999; · Zbl 0986.20048
[2] Detinko A.S., Maximal periodic subgroups of classical groups over fields of positive characteristic I, Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat., 1993, 4, 35-39 (in Russian); · Zbl 0925.20049
[3] Detinko A.S., Maximal periodic subgroups of classical groups over fields of positive characteristic II, Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat., 1994, 2, 49-52 (in Russian); · Zbl 0868.20036
[4] Detinko A.S., On deciding finiteness for matrix groups over fields of positive characteristic, LMS J. Comput. Math., 2001, 4, 64-72; · Zbl 1053.20041
[5] Dixon J.D., Zalesskii A.E., Finite primitive linear groups of prime degree, J. London Math. Soc., 1998, 57, 126-134 http://dx.doi.org/10.1112/S0024610798005778; · Zbl 0954.20012
[6] Dixon J.D., Zalesskii A.E., Finite imprimitive linear groups of prime degree, J. Algebra, 2004, 276, 340-370 http://dx.doi.org/10.1016/j.jalgebra.2004.02.005; · Zbl 1062.20051
[7] Flannery D.L., Detinko A.S., Locally nilpotent linear groups, Irish Math. Soc. Bull., 2005, 56, 37-51; · Zbl 1126.20032
[8] Isaacs I.M., Character theory of finite groups, Dover Publications, Inc., New York, 1994; · Zbl 0849.20004
[9] Konyukh V.S., Metabelian subgroups of the general linear group over an arbitrary field, Dokl. Akad. Nauk BSSR, 1978, 22, 389-392 (in Russian);
[10] Konyukh V.S., Sylow p-subgroups of a projective linear group, Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat., 1985, 6, 23-29 (in Russian); · Zbl 0644.20028
[11] Mazurova V.N., Maximal periodic subgroups of a symplectic group, Dokl. Akad. Nauk BSSR, 1985, 29, 403-406 (in Russian); · Zbl 0571.20041
[12] Mazurova V.N., Periodic subgroups of classical groups over fields of positive characteristic, Dokl. Akad. Nauk BSSR, 1985, 29, 493-496 (in Russian); · Zbl 0571.20042
[13] Suprunenko D.A., Matrix groups, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1976, 45;
[14] Wehrfritz B.A.F., Infinite linear groups, Springer-Verlag, Berlin, Heidelberg, New York, 1973; · Zbl 0261.20038
[15] Winter D.J., Representations of locally finite groups, Bull. Amer. Math. Soc., 1968, 74, 145-148 http://dx.doi.org/10.1090/S0002-9904-1968-11913-5; · Zbl 0159.31304
[16] Zalesskii A.E., Maximal periodic subgroups of the full linear group over a field with positive characteristic, Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk, 1966, 2, 121-123 (in Russian);
[17] Zalesskii A.E., Mazurova V.N., Maximal periodic subgroups of the orthogonal group, Institute of Mathematics AN BSSR, 1985, 9, 218 (in Russian);
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