On the soluble length of groups with prime-power order

Abstract

We show that for every integer k ≥ 3 and every prime p ≤ 5 there is a group with soluble length k and order .

Solvable groups with a given solvable length, and minimal composition length

“…solvable group) with solvable length d. Burnside [1] knew that c N (0) = 0, c N (1) = 1, c N (2) = 3, and c N (3) = 6. It is shown in [3] and [4] [12] and Schneider [15] further improved the lower bound to 2 d−1 +2d−4 and 2 d−1 +3d−10 respectively. Upper bounds are proved by producing specific examples.…”

The derived series of a finite p-group

“…Blackburn in his thesis [1] showed that if G is a p-group with odd order, then G (3) = 1 implies |G| p 14 . This bound was proved to be sharp in [7,8] (the show that in the special cases that I consider, the associated Lie rings preserve enough information about the derived series, and so they can be used to simplify the technical arguments.…”

The shape of solvable groups with odd order

“…The notation β p (d) is used in [2] to denote the minimal composition length of a pgroup with solvable length d. It is shown in [1] that β p (4) = 14 for p 5, and in [2] that β p (d) 2 d − 2 for p 5. The best known bounds for c N (d) are presently…”