# Mal'cev basis of a group

From Groupprops

## Definition

Let be a group. A **Mal'cev basis** of is a sequence of elements , such that, for every , the following are true:

- There exist such that
- The s are uniquely determined by , modulo the order of . In other words, if:

Then for every :

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.