Mal'cev basis of a group

Definition

Let $G$ be a group. A Mal'cev basis of $G$ is a sequence of elements $a_{1}, a_{2}, \dots, a_{n} \in G$ , such that, for every $g \in G$ , the following are true:

There exist $i_{1}, i_{2}, \dots, i_{n}$ such that $g = a_{1}^{i_{1}} a_{2}^{i_{2}} \dots a_{n}^{i_{n}}$
The $i_{j}$ s are uniquely determined by $g$ , modulo the order of $a_{j}$ . In other words, if:

$a_{1}^{i_{1}} a_{2}^{i_{2}} \dots a_{n}^{i_{n}} = a_{1}^{l_{1}} a_{2}^{l_{2}} \dots a_{n}^{l_{n}}$

Then for every $j$ :

$a_{j}^{i_{j} - l_{j}} = e$

A group possesses a Mal'cev basis if and only if it is polycyclic, and a Mal'cev basis can be used to construct a subnormal series with cyclic quotients. Conversely, given a subnormal series with cyclic quotients, we can pick representatives of generators for each factor, to obtain a Mal'cev basis.