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Applications of Lie rings with finite cyclic grading

Published online by Cambridge University Press:  05 July 2011

E. I. Khukhro
Affiliation:
Sobolev Institute of Mathematics
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Print publication year: 2011

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References

[1] J. L., Alperin, Automorphisms of solvable groups, Proc. Amer. Math. Soc. 13 (1962), 175–180.Google Scholar
[2] Yu. A., Bahturin and M. V., Zaicev, Identities of graded algebras, J. Algebra 205 (1998), 1–12.Google Scholar
[3] G. M., Bergman and I. M., Isaacs, Rings with fixed-point-free group actions, Proc. London Math. Soc. 27 (1973), 69–87.Google Scholar
[4] A., Borel and G. D., Mostow, On semi-simple automorphisms of Lie algebras, Ann. Math. (2) 61 (1955), 389–405.Google Scholar
[5] B., Bruno and F., Napolitani, A note on nilpotent-by-Černikov groups, Glasgow Math. J. 46 (2004) 211–215Google Scholar
[6] E. C., Dade, Carter subgroups and Fitting heights of finite solvable groups, Illinois J. Math. 13 (1969), 449–514.Google Scholar
[7] S., Donkin, Space groups and groups of prime power order. VIII. Pro-p-groups of finite coclass and p-adic Lie algebras, J. Algebra 111 (1987), 316–342.Google Scholar
[8] P., Fong, On orders of finite groups and centralizers of p-elements, Osaka J. Math. 13 (1976), 483–489.Google Scholar
[9] B., Hartley, A general Brauer–Fowler theorem and centralizers in locally finite groups, Pacific J. Math. 152 (1992), 101–117.Google Scholar
[10] B., Hartley and I. M., Isaacs, On characters and fixed points of coprime operator groups, J. Algebra 131 (1990), 342–358.Google Scholar
[11] B., Hartley and T., Meixner, Periodic groups in which the centralizer of an involution has bounded order, J. Algebra 64 (1980), 285–291.Google Scholar
[12] B., Hartley and T., Meixner, Finite soluble groups containing an element of prime order whose centralizer is small, Arch. Math. (Basel) 36 (1981), 211–213.Google Scholar
[13] B., Hartley and V., Turau, Finite soluble groups admitting an automorphism of prime power order with few fixed points, Math. Proc. Cambridge Philos. Soc. 102 (1987), 431–441.Google Scholar
[14] G., Higman, Groups and rings which have automorphisms without non-trivial fixed elements, J. London Math. Soc. (2) 32 (1957), 321–334.Google Scholar
[15] N., Jacobson, A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc. 6 (1955), 281–283.Google Scholar
[16] A., Jaikin-Zapirain, On almost regular automorphisms of finite p-groups, Adv. Math. 153 (2000), 391–402.Google Scholar
[17] A., Jaikin-Zapirain, Finite groups of bounded rank with an almost regular automorphism, Israel J. Math. 129 (2002), 209–220.Google Scholar
[18] E. I., Khukhro, Finite p-groups admitting an automorphism of order p with a small number of fixed points, Mat. Zametki 38 (1985), 652–657 (Russian); English transl., Math. Notes.38 (1986), 867–870.Google Scholar
[19] E. I., Khukhro, Groups and Lie rings admitting an almost regular automorphism of prime order, Mat. Sbornik 181, no. 9 (1990), 1207–1219; English transl., Math. USSR Sbornik71, no. 9 (1992), 51–63.Google Scholar
[20] E. I., Khukhro, Finite p-groups admitting p-automorphisms with few fixed points, Mat. Sb. 184, no. 12 (1993), 53–64; English transl., Russ. Acad. Sci., Sb., Math.80 (1995), 435–444.Google Scholar
[21] E. I., Khukhro, Almost regular automorphisms of finite groups of bounded rank, Sibirsk. Mat. Zh. 37 (1996), 1407–1412; English transl., Siberian Math. J.37 (1996), 1237–1241.Google Scholar
[22] E. I., Khukhro, On the solvability of Lie rings with an automorphism of finite order, Sibirsk. Mat. Zh. 42 (2001), 1187–1192); Siberian Math. J.42 (2001) 996–1000.Google Scholar
[23] E. I., Khukhro, Finite groups of bounded rank with an almost regular automorphism of prime order, Sibirsk. Mat. Zh. 43 (2002), 1182–1191; English transl., Siberian Math. J.43 (2002), 955–962.Google Scholar
[24] E. I., Khukhro, Groups with an automorphism of prime order that is almost regular in the sense of rank, J. London Math. Soc. 77 (2008), 130–148.Google Scholar
[25] E. I., Khukhro, Graded Lie rings with many commuting components and an application to 2-Frobenius groups, Bull. London Math. Soc. 40, (2008), 907–912.Google Scholar
[26] E. I., Khukhro, Lie rings with a finite cyclic grading in which there are many commuting components, Siberian Electron. Math. Rep. (http://semr.math.nsc.ru) 6 (2009), 243–250 (Russian).Google Scholar
[27] E. I., Khukhro, Ant. A., Klyachko, N. Yu., Makarenko, Yu. B., Melnikova, Automorphism invariance and identities, Bull. London Math. Soc. 41 (2009), 804–816.Google Scholar
[28] E. I., Khukhro and N. Yu., Makarenko, Large characteristic subgroups satisfying multilinear commutator identities, J. London Math. Soc. 75, no. 3 (2007), 635–646.Google Scholar
[29] E. I., Khukhro and N. Yu., Makarenko, Automorphically-invariant ideals satisfying multilinear identities, and group-theoretic applications, J. Algebra 320 (2008), 1723–1740.Google Scholar
[30] E. I., Khukhro, N. Yu., Makarenko, and P., Shumyatsky, Nilpotent ideals in graded Lie algebras and almost constant-free derivations, Commun. Algebra 36 (2008), 1869–1882.Google Scholar
[31] E. I., Khukhro and V. D., Mazurov, Finite groups with an automorphism of prime order whose centralizer has small rank, J. Algebra 301 (2006), 474–492.Google Scholar
[32] E. I., Khukhro and V. D., Mazurov, Automorphisms with centralizers of small rank, in Groups St Andrews 2005, Vol. 2 (C. M., Campbell et al., eds.), London Math. Soc. Lecture Note Ser. 340 (CUP, Cambridge 2007), 564–585.Google Scholar
[33] E. I., Khukhro and P. V., Shumyatsky, On fixed points of automorphisms of Lie rings and locally finite groups, Algebra Logika 34 (1995), 706–723; English transl., Algebra Logic34 (1995), 395–405.Google Scholar
[34] E. I., Khukhro and P., Shumyatsky, Nilpotency of finite groups with Frobenius groups of automorphisms, submitted, 2009.Google Scholar
[35] I., Kiming, Structure and derived length of finite p-groups possessing an automorphism of p-power order having exactly p fixed points, Math. Scand. 62 (1988), 153–172.Google Scholar
[36] L. G., Kovacs, Groups with regular automorphisms of order four, Math. Z. 75 (1961), 277–294.Google Scholar
[37] V. A., Kreknin, The solubility of Lie algebras with regular automorphisms of finite period, Dokl. Akad. Nauk SSSR 150 (1963), 467–469; English transl., Math. USSR Doklady4 (1963), 683–685.Google Scholar
[38] V. A., Kreknin, Solvability of a Lie algebra containing a regular automorphism, Sibirsk. Mat. Zh. 8 (1967), 715–716; English transl., Siberian Math. J.8 (1967), 536–537.Google Scholar
[39] V. A., Kreknin and A. I., Kostrikin, Lie algebras with regular automorphisms, Dokl. Akad. Nauk SSSR 149 (1963), 249–251; English transl., Math. USSR Doklady4 (1963), 355–358.Google Scholar
[40] C. R., Leedham-Green, Pro-p-groups of finite coclass, J. London Math. Soc. 50 (1994), 43–48.Google Scholar
[41] C. R., Leedham-Green, The structure of finite p-groups, J. London Math. Soc. 50 (1994), 49–67.Google Scholar
[42] C. R., Leedham-Green and S., McKay, On p-groups of maximal class. I, Quart. J. Math. Oxford Ser. 27 (1976), 297–311; II, ibid.29 (1978), 175–186; III, ibid.29 (1978), 281–299.Google Scholar
[43] C. R., Leedham-Green, S., McKay, and W., Plesken, Space groups and groups of prime power order. V. A bound to the dimension of space groups with fixed coclass, Proc. London Math. Soc. 52 (1986), 73–94.Google Scholar
[44] C. R., Leedham-Green and M. F., Newman, Space groups and groups of prime power order. I, Arch. Math. (Basel) 35 (1980), 193–202.Google Scholar
[45] V., Linchenko, Identities of Lie algebras with actions of Hopf algebras, Commun. Algebra 25 (1997) 3179–3187; erratum, ibid.31 (2003), 1045–1046.Google Scholar
[46] N. Yu., Makarenko, On almost regular automorphisms of prime order, Sib. Matem. Zh. 33, no. 5 (1992), 206–208; English transl., Sib. Math. J.33, no. 5 (1992), 932–934.Google Scholar
[47] N. Yu., Makarenko, Finite 2-groups with automorphisms of order 4, Algebra Logika 40 (2001), 83–96; English transl., Algebra Logic40 (2001), 47–54.Google Scholar
[48] N. Yu., Makarenko, A nilpotent ideal in the Lie rings with an automorphism of prime order, Sibirsk. Mat. Zh. 46 (2005), 1361–1374; English transl., Siberian Math. J.46 (2005), 1097–1107.Google Scholar
[49] N. Yu., Makarenko, Small centralizers in groups and Lie rings, Diss. … Doktor Fiz.-Mat. Nauk (Inst. Math., Novosibirsk 2006) (Russian).
[50] N. Yu., Makarenko, Graded Lie algebras with a few non-trivial components, Sibirsk. Mat. Zh. 48 (2007), 116–137; English transl., Siberian Math. J.48 (2007), 95–111.Google Scholar
[51] N. Yu., Makarenko and E. I., Khukhro, Nilpotent groups admitting an almost regular automorphism of order 4, Algebra Logika 35 (1996), 314–333; English transl., Algebra Logic35 (1996), 176–187.Google Scholar
[52] N. Yu., Makarenko and E. I., Khukhro, Lie rings admitting an automorphism of order 4 with few fixed points. II, Algebra Logika 37 (1998), 144–166; English transl., Algebra Logic37, no. 2 (1998), 78–91.Google Scholar
[53] N. Yu., Makarenko and E. I., Khukhro, Almost solubility of Lie algebras with almost regular automorphisms, J. Algebra 277 (2004), 370–407.Google Scholar
[54] N. Yu., Makarenko and E. I., Khukhro, Finite groups with an almost regular automorphism of order four, Algebra Logika 45 (2006), 575–602; English transl., Algebra Logic45 (2006), 326–343.Google Scholar
[55] N. Yu., Makarenko and P., Shumyatsky, Frobenius groups as groups of automorphisms, Preprint (Brazilia–Mulhouse 2009).Google Scholar
[56] V. D., Mazurov, Recognition of the finite simple groups S4(q) by their element orders, Algebra Logika 41 (2002), 166–198; English transl., Algebra Logic41 (2002), 93–110.Google Scholar
[57] S., McKay, On the structure of a special class of p-groups, Quart. J. Math. Oxford 38 (1987), 489–502.Google Scholar
[58] Yu., Medvedev, Groups and Lie rings with almost regular automorphisms, J. Algebra 164 (1994), 877–885.Google Scholar
[59] Yu., Medvedev, p-Divided Lie rings and p-groups, J. London Math. Soc. 59 (1999), 787–798.Google Scholar
[60] M. R., Pettet, Automorphisms and Fitting factors of finite groups, J. Algebra 72 (1981), 404–412.Google Scholar
[61] P., Rowley, Finite groups admitting a fixed-point-free automorphism group, J. Algebra 174 (1995), 724–727.Google Scholar
[62] A., Shalev, Automorphisms of finite groups of bounded rank, Israel J. Math. 82 (1993), 395–404.Google Scholar
[63] A., Shalev, On almost fixed point free automorphisms, J. Algebra 157 (1993), 271–282.Google Scholar
[64] A., Shalev, The structure of finite p-groups: effective proof of the coclass conjectures, Invent. Math. 115 (1994), 315–345.Google Scholar
[65] A., Shalev and E. I., Zelmanov, Pro-p-groups of finite coclass, Math. Proc. Cambridge Philos. Soc. 111 (1992), 417–421.Google Scholar
[66] R., Shepherd, p-Groups of maximal class, Ph. D. Thesis (Univ. of Chicago 1971).
[67] P., Shumyatsky, Involutory automorphisms of finite groups and their centralizers, Arch. Math. (Basel) 71 (1998), 425–432.Google Scholar
[68] P., Shumyatsky, On locally finite groups and the centralizers of automorphisms, Boll. Unione Mat. Italiana 4 (2001), 731–736.Google Scholar
[69] P., Shumyatsky, Finite groups and the fixed points of coprime automorphisms, Proc. Amer. Math. Soc. 129 (2001), 3479–3484.Google Scholar
[70] V. P., Shunkov, On periodic groups with an almost regular involution, Algebra Logika 11 (1972), 470–493; English transl., Algebra Logic11 (1973), 260–272.Google Scholar
[71] J., Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 578–581.Google Scholar
[72] J., Thompson, Automorphisms of solvable groups, J. Algebra 1 (1964), 259–267.Google Scholar
[73] A., Turull, Fitting height of groups and of fixed points, J. Algebra 86 (1984), 555–566.Google Scholar
[74] D. J., Winter, On groups of automorphisms of Lie algebras, J. Algebra 8 (1968), 131–142.Google Scholar

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