Geodesic
Groups with finitely many conjugacy classes of subgroups with large subnormal defect
Glasgow mathematical journal, Tome 37 (1995) no. 1, pp. 69-71

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Given a group G and a positive integer k, let vk(G) denote the number of conjugacy classes of subgroups of G which are not subnormal of defect at most k. Groups G such that vkG) < ∝ for some k are considered in Section 2 of [1], and Theorem 2.4 of that paper states that an infinite group G for which vk(G) < ∝ (for some k) is nilpotent provided only that all chief factors of G are locally (soluble or finite). Now it is easy to see that a group G whose chief factors are of this type is locally graded, that is, every nontrivial, finitely generated subgroup F of G has a nontrivial finite image (since there is a chief factor H/K of G such that F is contained in H but not in K). On the other hand, every (locally) free group is locally graded and so there is in general no restriction on the chief factors of such groups. The class of locally graded groups is a suitable class to consider if one wishes to do no more than exclude the occurrence of finitely generated, infinite simple groups and, in particular, Tarski p-groups. As pointed out in [1], Ivanov and Ol'shanskiĭ have constructed (finitely generated) infinite simple groups all of whose proper nontrivial subgroups are conjugate; clearly a group G with this property satisfies v1(G) = l. The purpose of this note is to provide the following generalization of the above-mentioned theorem from [1]. 
Smith, Howard. Groups with finitely many conjugacy classes of subgroups with large subnormal defect. Glasgow mathematical journal, Tome 37 (1995) no. 1, pp. 69-71. doi: 10.1017/S0017089500030408
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[1] 1.Brandl, R., Franciosi, S., Giovanni, F. de, Groups with finitely many conjugacy classes of non-normal subgroups, (to appear). Google Scholar

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

[3] 3.Roseblade, J. E., On groups in which every subgroup is subnormal, J. Algebra 2 (1965), 402–412. Google Scholar

[4] 4.Zel'manov, E. I., Solution of the restricted Burnside problem for groups of odd exponent, Izv. Akad. Nauk. SSR Ser. Mat. 54 (1990), 42–59. Google Scholar

[5] 5.Zel'manov, E. I., Solution of the restricted Burnside problem for 2-groups., Mat. Sb. 182 (1991), 568–592. Google Scholar

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