In this package we compute the canonical conjugacy representatives of elements and subgroups of nilpotent groups given by a nilpotent sequence. This package also includes functions to compute the intersection of subgroups of nilpotent groups and to compute the product of subgroups of nilpotent groups.
Let G be a finitely generated nilpotent group. Then G has a series G = G_1 > G_2 > \dots > G_n > G_{n+1} = \{1\} so that each G_i is normal in G and each quotient G_i/G_{i+1} is cyclic and central in G/G_{i+1}. We call such a series a nilpotent series of G. Then (g_1,\dots, g_n) is called a nilpotent sequence of G and (o_1,\dots, o_n) are its relative orders. Let I=\{ i\in \mathbb{Z} \mid o_i \neq \infty \}. Note that the nilpotent sequence determines its nilpotent series by G_i = \langle g_i,\dots, g_n \rangle holds for 1 \leq i \leq n.
Each element g \in G can be written uniquely as g = g_1^{e_1} \dots g_n^{e_n} with e_i \in \mathbb{Z} and e_i \in \{0,\dots, o_{i}-1\} if i \in I. The factorisation g_1^{e_1} \dots g_n^{e_n} for g \in G is called the normal form of g. The associated integer vector (e_1,\dots, e_n) is the exponent vector of g. We write e(g) = (e_1, \dots , e_n). If e_1 = \dots = e_{i−1} = 0 and e_i \neq 0, then we write \mathrm{dep}(g) = i and call this the depth of g. The leading exponent of an element g is e_d where d = \mathrm{dep}(g). The identity element satisfies \mathrm{dep}(1) = n + 1 and does not have leading exponent.
generated by GAPDoc2HTML