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:


 * 1) There exist $$i_1, i_2, \dots, i_n$$ such that $$g = a_1^{i_1}a_2^{i_2}\dots a_n^{i_n}$$
 * 2) 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.