# 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.