Finitely generated soluble groups in which all subgroups have finite lower central depth. (English) Zbl 0314.20029

Lennox, John C.

Bull. Lond. Math. Soc. 7, 273-278 (1975).

A group \( G \) is said to have finite (lower central) depth if the lower central series of \( G \) stabilises after a finite number of steps. The main result of the paper is that a finitely generated soluble group has all of its subgroups of finite depth if and only if it is finite by nilpotent. This fact can be read off from either Theorem 1. A finitely generated soluble group in which each two generator subgroup has finite depth is finite by nilpotent, or from Theorem 2. A finitely generated soluble group in which each characteristic subgroup of finite index has finite depth is finite by nilpotent. - It is shown in Theorem 3 that Theorem 1 extends to the class of all finitely generated soluble by finite groups which fact in turn leads to the corollary that finitely generated linear groups in which all two generator subgroups have finite depth are finite by nilpotent. An example shows that Theorem 2 cannot even be extended to the class of finitely generated abelian by finite groups.

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MSC:

20E15

Chains and lattices of subgroups, subnormal subgroups