Nilpotent groups related to an automorphism. (English) Zbl 1448.20031
Summary: The aim of this paper is to state some results on an \(\alpha\)-nilpotent group, which was recently introduced by R. Barzegar and A. Erfanian [Casp. J. Math. Sci. 4, No. 2, 271–283 (2015; Zbl 1424.20038)], for any fixed automorphism \(\alpha\) of a group \(G\). We define an identity nilpotent group and classify all finitely generated identity nilpotent groups. Moreover, we prove a theorem on a generalization of the converse of the known Schur’s theorem. In the last section of the paper, we study absolute normal subgroups of a finite group.
MSC:
20F18 | Nilpotent groups |
20D15 | Finite nilpotent groups, \(p\)-groups |
20F12 | Commutator calculus |
20D45 | Automorphisms of abstract finite groups |
20E36 | Automorphisms of infinite groups |
Citations:
Zbl 1424.20038References:
[1] | Abdollahi, A., Finite \(p\)-groups of class \(2\) have non-inner automorphisms of order \(p\), J. Algebra, 312, 876-879, (2007) · Zbl 1120.20024 · doi:10.1016/j.jalgebra.2006.08.036 |
[2] | Barzegar, R.; Erfanian, A., Nilpotency and solubility of groups relative to an automorphism, Caspian J. Math. Sci., 4, 271-283, (2015) · Zbl 1424.20038 |
[3] | Christopher, JH; Darren, LR, Automorphism of finite abelian groups, Am. Math. Mon., 114, 917-923, (2007) · Zbl 1156.20046 · doi:10.1080/00029890.2007.11920485 |
[4] | Deaconescu, M.; Silberberg, G., Non-inner automorphisms of order \(p\) of finite \(p\)-groups, J. Algebra, 250, 283-287, (2002) · Zbl 1012.20017 · doi:10.1006/jabr.2001.9093 |
[5] | Ganjali, M.; Erfanian, A., Perfect groups and normal subgroups related to an automorphism, Ricerche Mat., 66, 407-413, (2017) · Zbl 1377.20018 · doi:10.1007/s11587-016-0307-7 |
[6] | Garrison, D.; Kappe, L-C; Yull, D., Autocommutators and the autocommutator subgroup, Contemp. Math., 421, 137-146, (2006) · Zbl 1156.20029 · doi:10.1090/conm/421/08033 |
[7] | Gaschütz, W., Nichtabelsche \(p\)-gruppen besitzen äussere \(p\)-automorphismen, J. Algebra, 4, 1-2, (1966) · Zbl 0142.26001 · doi:10.1016/0021-8693(66)90045-7 |
[8] | Hegarty, P., The absolute centre of a group, J. Algebra, 169, 929-935, (1994) · Zbl 0817.20037 · doi:10.1006/jabr.1994.1318 |
[9] | Khukhro E I, Nilpotent groups and their automorphisms (1993) (Berlin: Walter de Gruyter) · Zbl 0795.20018 · doi:10.1515/9783110846218 |
[10] | Miller, GA, Determination of all the groups of order \(p^{m}\) which contain the abelian group of type \((m-2, 1)\), \(p\) being any prime, Tran. AMS, 2, 259-272, (1901) · JFM 32.0143.02 |
[11] | Miller, GA, On the groups of order \(p^m\) which contain operators of order \(p^{m-2}\), Trans. AMS, 3, 383-387, (1902) · JFM 33.0154.01 |
[12] | Niroomand, P., The converse of Schur’s theorem, Arch. Math., 94, 401-403, (2010) · Zbl 1197.20023 · doi:10.1007/s00013-010-0106-4 |
[13] | Parvaneh, F., Some properties of autonilpotent groups, Italian J. Pure Appl. Math., 35, 1-8, (2015) · Zbl 1338.20022 |
[14] | Robinson D J S, A course in the theory of groups, Graduate Texts in Mathematics 80 (1996) (New York: Springer) |
[15] | Winter, DL, The automorphism group of an extraspecial \(p\)-group, Rocky Mountain J. Math., 2, 159-168, (1972) · Zbl 0242.20023 · doi:10.1216/RMJ-1972-2-2-159 |
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