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.
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