<p>In this package we demonstrate the algorithmic usefulness of the so-called Mal'cev correspondence for computations with infinite polycyclic groups; it is a correspondence that associates to every Q-powered nilpotent group H a unique rational nilpotent Lie algebra L_H and vice-versa. The Mal'cev correspondence was discovered by Anatoly Mal'cev in 1951 [Mal51].
<h4>1.2 <span class="Heading">Setup for computing the correspondence</span></h4>
<p>Let <span class="SimpleMath">G</span> be a finitely generated torsion-free nilpotent group, i.e.\ a <span class="SimpleMath">T</span>-group. Then <span class="SimpleMath">G</span> can be embedded in a <span class="SimpleMath">Q</span>-powered hull <span class="SimpleMath">G^</span>. The group <span class="SimpleMath">G^</span> is a <span class="SimpleMath">Q</span>-powered nilpotent group and is unique up to isomorphism. We denote the Lie algebra which corresponds to <span class="SimpleMath">G^</span> under the Mal'cev correspondence by L(G)= L_G^. We provide an algorithm for setting up the Mal'cev correspondence between <span class="SimpleMath">G^</span> and the Lie algebra <span class="SimpleMath">L(G)</span>. That is, if <span class="SimpleMath">G</span> is given by a polycyclic presentation with respect to a Mal'cev basis, then we can compute a structure constants table of L(G). Furthermore for a given g∈ G we can compute the corresponding element in L(G) and vice versa.
<p>Every element of a polycyclically presented group has a unique normal form. An algorithm for computing this normal form is called a collection algorithm. Such an algorithm lies at the heart of most methods dealing with polycyclically presented groups. The current state of the art is collection from the left <a href="chapBib.html#biBGeb02">[Geb02]</a><a href="chapBib.html#biBLGS90">[LGS90]</a><a href="chapBib.html#biBVLe90">[VL90]</a> }. This package contains a new collection algorithm for polycyclically presented groups, which we call Mal'cev collection [AL07]. Mal'cev collection is in some cases dramatically faster than collection from the left, while using less memory.</p>
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