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On groups with two isomorphism classes of derived subgroups. (English) Zbl 1287.20046
If \(H\) is a subgroup of a group \(G\) then \(H\) is a derived subgroup of \(G\) if \(H=K'\) for some subgroup \(K\) of \(G\). Let \(\mathfrak C_n\) denote the class of groups in which there are at most \(n\) derived subgroups and let \(\mathfrak C\) denote the union of the classes \(\mathfrak C_n\). These classes have recently been investigated by F. de Giovanni and D. J. S. Robinson, [J. Lond. Math. Soc., II. Ser. 71, No. 3, 658-668 (2005; Zbl 1084.20026)], and M. Herzog, P. Longobardi and M. Maj [Contemp. Math. 402, Isr. Math. Conf. Proc. 181-192 (2006; Zbl 1122.20017)]. By contrast this paper is concerned with the groups in which the set of isomorphism types of derived subgroups is small.
If \(n\) is a positive integer, let \(\mathfrak D_n\) denote the class of groups whose derived subgroups fall into at most \(n\) isomorphism classes. Obviously, \(\mathfrak D_1=\mathfrak C_1\) is the class of Abelian groups and in this paper the class \(\mathfrak D_2\) is discussed. Thus a non-Abelian group \(G\in\mathfrak D_2\) if and only if \(H'\cong G'\) whenever \(H\) is a non-Abelian subgroup of \(G\). There are many diverse examples of \(\mathfrak D_2\)-groups. Among the many interesting theorems we give the flavor with two of these: Theorem 1: A non-Abelian group \(G\) is nilpotent and belongs to \(\mathfrak D_2\) if and only if \(G'\) is cyclic of prime or infinite order and \(G'\leq Z(G)\); Theorem 3 (abridged): Let \(G\) be a non-nilpotent soluble \(\mathfrak D_2\)-group. Then \(G\) is metabelian; \(G'\) is an elementary Abelian \(p\)-group for some prime \(p\), or a free Abelian group, or a torsion-free minimax group; and the nilpotent subgroups of \(G\) are Abelian.

20F14 Derived series, central series, and generalizations for groups
20E34 General structure theorems for groups
20F16 Solvable groups, supersolvable groups
20F18 Nilpotent groups
Full Text: DOI
[1] DOI: 10.1016/0022-4049(75)90004-3 · Zbl 0311.20012
[2] DOI: 10.1090/S0002-9947-1903-1500650-9
[3] DOI: 10.1093/acprof:oso/9780198507284.001.0001
[4] DOI: 10.1090/conm/402/07578
[5] Robinson, Symposia Math. 17 pp 441– (1976)
[6] DOI: 10.1112/S0024610705006484 · Zbl 1084.20026
[7] DOI: 10.1007/s00013-005-1567-8 · Zbl 1100.20030
[8] DOI: 10.1007/BF01350807 · Zbl 0183.02602
[9] DOI: 10.1016/0021-8693(72)90058-0 · Zbl 0236.20032
[10] Granville, Trans. Amer. Math. Soc. 306 pp 329– (1988)
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