Geodesic
Groups isomorphic to their non-nilpotent subgroups
Glasgow mathematical journal, Tome 40 (1998) no. 2, pp. 257-262

Voir la notice de l'article provenant de la source Cambridge University Press

We were concerned in [12] with groups G that are isomorphic to all of their non-abelian subgroups. In order to exclude groups with all proper subgroups abelian, which are well understood in the finite case [7] and which include Tarski groups in the infinite case, we restricted attention to the class X of groups G that are isomorphic to their nonabelian subgroups and that contain proper subgroups of this type; such groups are easily seen to be 2-generator, and a complete classification was given in [12, Theorem 2] for the case G soluble. In the insoluble case, G/Z(G) is infinite simple [12; Theorem 1], though not much else was said in [12] about such groups. Here we examine a property which represents a natural generalisation of that discussed above. Let us say that a group G belongs to the class W if G is isomorphic to each of its non-nilpotent subgroups and not every proper subgroup of G is nilpotent. Firstly, note that finite groups in which all proper subgroups are nilpotent are (again) well understood [9]. In addition, much is known about infinite groups with all proper subgroups nilpotent (see, in particular, [8] and [13] for further discussion) although, even in the locally nilpotent case, there are still some gaps in our understanding of such groups. We content ourselvesin the present paper with discussing finitely generated W-groups— note that a W-group is certainly finitely generated or locally nilpotent. We shall have a little more to say about the locally nilpotent case below. 
Smith, Howard; Wiegold, James. Groups isomorphic to their non-nilpotent subgroups. Glasgow mathematical journal, Tome 40 (1998) no. 2, pp. 257-262. doi: 10.1017/S0017089500032572
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[1] 1.Gruenberg, K. W., Two theorems on Engel groups, Proc. Camb. Phil. Soc. 49 (1953), 377–380. Google Scholar | DOI

[2] 2.Hall, P., Finiteness conditions for soluble groups, Proc. London Math. Soc. 4 (1954), 419–436. Google Scholar

[3] 3.Hall, P., The Edmonton notes on nilpotent groups, QMC notes, London 1969. Google Scholar

[4] 4.Kropholler, P., On finitely generated soluble groups with no large wreath product sections, Proc. London Math. Soc. 49 (1984), 155–169. Google Scholar | DOI

[5] 5.Lennox, J. C., Smith, H. and Wiegold, J., A problem on normal subgroups, J. Pure and Applied Alg. 88 (1993), 169–171. Google Scholar | DOI

[6] 6.Longobardi, P., Maj, M. and Smith, H., A note on locally graded groups, Rend. Sem. Mat. Univ. Padova 94 (1995), 275–277. Google Scholar

[7] 7.Miller, G. A. and Moreno, H. C., Non-abelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc. 4 (1903), 389–404. Google Scholar | DOI

[8] 8.Newman, M. F. and Wiegold, J., Groups with many nilpotent subgroups, Archiv der Math. 15 (1964), 241–250. Google Scholar | DOI

[9] 9.Rédei, L., Die endlichen einstufig nichtnilpotenten Gruppen, Publ. Math. Debrecen 4 (1956), 130–138. Google Scholar

[10] 10.Robinson, D. J. S., Finiteness conditions and generalized soluble groups (2 vols.), (Springer, Berlin-Heidelberg-NewYork, 1972). Google Scholar

[11] 11.Robinson, D. J. S., A course in the theory of groups (Springer, Berling-Heidelberg-New York, 1982). Google Scholar | DOI

[12] 12.Smith, H. and Wiegold, J., Groups which are isomorphic to their nonabelian subgroups, Rend. Sem. Mat. Univ. Paclova 97 (1997), 7–16. Google Scholar

[13] 13.Smith, H., Groups with few non-nilpotent subgroups, Glasgow Math. J. 39 (1997), 141–151. Google Scholar | DOI

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