Mal'cev basis of a group
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.