Mal'cev basis of a group

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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_js 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.