Let G be a polycyclic group with a pc series as above. A polycyclic generating sequence (pcgs for short) of G is a sequence P : = (g1, ¼, gn) of elements of G such that Ci = áCi+1, gi ñ for 1 £ i £ n. Note that each polycyclic group has a pcgs, but except for very small groups, a pcgs is not unique.
For each index i the subsequence of elements (gi, ¼, gn) forms a pcgs of the subgroup Ci. In particular, these tails generate the subgroups of the pc series and hence we say that the pc series is determined by P.
Let ri be the index of Ci+1 in Ci which is either a finite positive number or infinity. Then ri is the order of gi Ci+1 and we call the resulting list of indices the relative orders of the pcgs P.
Moreover, with respect to a given pcgs (g1, ¼, gn) each element g of G can be represented in a unique way as a product g = g1e1 ·g2e2 ¼gnen with exponents ei Î {0, ¼, ri-1}, if ri is finite, and ei Î Z otherwise. Words of this form are called normal words or words in normal form. Then the integer vector [e1, ¼, en] is called the exponent vector of the element g. Furthermore, the smallest index k such that ek ¹ 0 is called the depth of g and ek is the leading exponent of g.
For many applications we have to assume that each of the relative orders ri is either a prime or infinity. This is equivalent to saying that there are no trivial factors in the pc series and the finite factors of the pc series are maximal refined. Then we obtain that ri is the order of g Ci+1 for all elements g in Ci \Ci+1 and we call ri the relative order of the element g.
GAP 4 manual