Poly-nilpotent group

A group possessing a finite normal series with nilpotent factors; such a series is called poly-nilpotent. The length of the shortest poly-nilpotent series of a poly-nilpotent group is called its poly-nilpotent length. The class of all poly-nilpotent groups coincides with the class of all solvable groups (cf. Solvable group); however, in general the poly-nilpotent length is less than the solvable length. A poly-nilpotent group of length 2 is called meta-nilpotent.

All groups having (an increasing) poly-nilpotent series of length $l$ whose factors in increasing order have nilpotent classes not exceeding the numbers $c_1,\dots,c_l$, respectively, form a variety $\mathfrak M$, which is the product of nilpotent varieties:

$$\mathfrak M=\mathfrak N_{c_1}\dots\mathfrak N_{c_l}$$

(see Variety of groups). The free groups of such a variety are called free poly-nilpotent groups. Of particular interest are the varieties $\mathfrak N_c\mathfrak A$ and $\mathfrak A\mathfrak N_c$. The first of them contains all connected solvable Lie groups; in the second, all finitely-generated groups are finitely approximable and satisfy the maximum condition for normal subgroups.

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