Subgroups of the upper-triangular matrix group with maximal derived length and a minimal number of generators

Abstract

The group Un(F) of all nxn unipotent upper-triangular matrices over F has derived length d := Ceiling(log2 (n)), equivalently 2d-1 < n <= 2d. We prove that Un(F) has a 3-generated subgroup of derived length d, and it has a 2-generated subgroup of derived length d if and only if (21/32)* 2d < n <= 2d.

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