<p>Let <span class="SimpleMath">G</span> be an infinite polycyclic group. It is well-known that there exist a normal <span class="SimpleMath">T</span>-group <span class="SimpleMath">N</span> and a <span class="SimpleMath">T</span>-group <span class="SimpleMath">C</span> such that <span class="SimpleMath">H=CN</span> is normal of finite index in <span class="SimpleMath">G</span> and <span class="SimpleMath">H/N</span> is free abelian of finite rank <a href="chapBib.html#biBSeg83">[Seg83]</a>. In this chapter we present an effective collection method for an infinite polycyclic group which is given by a polycyclic presentation with respect to a polycyclic sequence <span class="SimpleMath">P</span> going through the normal series <span class="SimpleMath">1 ≤ N ≤ H ≤ G</span>. This polycyclic sequence <span class="SimpleMath">P</span> must be chosen as follows. Let <span class="SimpleMath">(n_1,dots,n_l)</span> be a Mal'cev basis of N and let (c_1N,dots,c_k N) be a basis for the free abelian group CN/N. Then (c_1,dots,c_k,n_1,dots,n_l) is a polycyclic sequence for H=CN. Further there exists f_1,dots, f_j ∈ G such that (f_1 H, dots, f_j H) is a polycyclic sequence for G/H. Now we set
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MalcevCollectorConstruction</code>( <var class="Arg">G</var>, <var class="Arg">inds</var>, <var class="Arg">C</var>, <var class="Arg">CC</var>, <var class="Arg">N</var>, <var class="Arg">NN</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns a Mal'cev collector for the infinite polycyclically presented group G. The group G must be given with respect to a polycyclic sequence (g_1,dots,g_r, c_r+1, dots, c_r+s, n_r+s+1, dots, n_r+s+t) with the following properties:
<ul>
<li><p>(a) <span class="SimpleMath">(n_r+s+1, dots, n_r+s+t)</span> is a Mal'cev basis for the T-group N ≤ G,
</li>
<li><p>(b) <span class="SimpleMath">(c_r+1N, dots, c_r+sN)</span> is a basis for the free-abelian group <span class="SimpleMath">CN/N</span> where <span class="SimpleMath">C ≤ G</span> is a <span class="SimpleMath">T</span>-group generated by <span class="SimpleMath">c_r+1, dots, c_r+s</span>,</p>
</li>
<li><p>(c) <span class="SimpleMath">(g_1 CN, dots, g_r CN)</span> is a polycyclic sequence for the finite group <span class="SimpleMath">G/CN</span>.</p>
</li>
</ul>
<p>The list <var class="Arg">inds</var> is equal to <span class="SimpleMath">[ [1,dots,r],[r+1,dots,r+s],[r+s+1,dots,r+s+t]]</span>. The group <span class="SimpleMath">CC</span> is isomorphic to <span class="SimpleMath">C</span> via <var class="Arg">CC</var>!.bijection and given by a polycyclic presentation with respect to a Mal'cev basis starting with c_r+1, dots, c_r+s. The group NN is isomorphic to N via NN!.bijection. and given by a polycyclic presentation with respect to the Mal'cev basis <span class="SimpleMath">( n_r+s+1, dots, n_r+s+t)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GUARANA.Tr_n_O1</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GUARANA.Tr_n_O2</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>for a positive integer <var class="Arg">n</var> these functions construct polycyclically presented groups that can be used to test the Mal'cev collector. They return a list which can be used as input for the function MalcevCollectorConstruction. The constructed groups are isomorphic to triangular matrix groups of dimension n over the ring O_1, respectively O_2. The ring O_1, respectively O_2, is the maximal order of Q(θ_i) where θ_1, respectively θ_2, is a zero of the polynomial p_1(x) = x^2-3, respectively p_2(x)=x^3 -x^2 +4.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GUARANA.F_2c_Aut1</code>( <var class="Arg">c</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GUARANA.F_3c_Aut1</code>( <var class="Arg">c</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>for a positive integer <var class="Arg">c</var> these functions construct polycyclically presented groups that can be used to test the Mal'cev collector. They return a list which can be used as input for the function MalcevCollectorConstruction. These groups are constructed as follows: Let F_n,c be the free nilpotent of class c group on n generators. An automorphism φ of the free group F_n naturally induces an automorphismbarφ of F_n,c. We use the automorphism φ_1 of F_2 which maps f_1 to f_2^-1 and f_2 to f_1 f_2^3 and the automorphism φ_2 of F_3 mapping f_1 to f_2^-1, f_2 to f_3^-1 and f_3 to f_2^-3f_1^-1 for our construction. The returned group F_2c_Aut1, respectively F_3c_Aut2, is isomorphic to the semidirect product ⟨ φ_1 ⟩ ⋉ F_2,c, respectively ⟨ φ_2 ⟩ ⋉ F_3,c.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MalcevGElementByExponents</code>( <var class="Arg">malCol</var>, <var class="Arg">exps</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev collector malCol of a group G and an exponent vector exps with integer entries, this functions returns the group element of G, which has exponents exps with respect to the polycyclic sequence underlying malCol.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Random</code>( <var class="Arg">malCol</var>, <var class="Arg">range</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev collector malCol this function returns the output of MalcevGElementByExponents( malCol, exps ), where exps is an exponent vector whose entries are randomly chosen integers between -range and range.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ *</code>( <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the product of group elements which are defined with respect to a Mal'cev collector by the the function MalcevGElementByExponents.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GUARANA.AverageRuntimeCollec</code>( <var class="Arg">malCol</var>, <var class="Arg">ranges</var>, <var class="Arg">no</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev collector malCol, a list of positive integers ranges and a positive integer no this function computes for each number r in ranges the average runtime of no multiplications of two random elements of malCol of range r, as generated by Random( malCol, r ).
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