Suppose that the finitely generated group G is given by a nilpotent sequence (g_1,\dots,g_n). For g \in G and U \leq G, we denote the conjugacy class of g under U by g^U = \{g^h \mid h \in U\} and the centralizer of g in U by C_U(g) = \{h \in U \mid g^h = g\}.
Let \ll be the well-order 0 \ll 1 \ll 2 \ll \dots \ll −1 \ll −2 \dots of \mathbb{Z}. We extend the well-order to exponent vectors lexicographically; that is, (e_1,\dots, e_n) \ll (f_1, \dots, f_n) if e_1 = f_1, \dots , e_{i−1} = f_{i−1} and e_i \ll f_i for some i \in {1,\dots,n} . We order the elements of G via their exponent vectors; that is, g \ll h if e(g) \leq e(h). One of the main programs of this package is to compute the minimum element in g^U with respect to \ll; this element is known as the canonical conjugacy representative of g, which is denoted as \mathrm{Cano}_U(g). This program also computes the centralizer C_U(g). This package is supplementary to the article of Eick and Fernández Ayala [EF25].
‣ CentralizerNilGroup( G, elms ) | ( function ) |
Computes the centralizer of a given set of elements elms in G.
‣ IsConjugateNilGroup( G, g, h ) | ( function ) |
Checks if two given elements g and h in G are conjugate. If so it returns the conjugating element.
‣ CanonicalConjugateElements( G, elms ) | ( function ) |
Computes the canonical conjugate representative of a given elements elms in G. Returns a record containing the canonical conjugates, the conjugating elements to obtain the canonical conjugate and the centralizers of the given elements.
‣ IsCanonicalConjugateElements( G, elms ) | ( function ) |
Checks if a set of elements elms in G are conjugate using the canonical representative aproach. If so it returns a record containing the canonical conjugate representative, the conjugating elements and the centralizer.
‣ InfoConjugacyElements | ( info class ) |
Info class for the functions of canonical conjugates for elements.
Suppose that the finitely generated group G is given by a nilpotent sequence (g_1,\dots,g_n). Given U, V \leq G we write U^V = \{U^g \mid g ∈ V \} for the conjugacy class of U under V and N_V(U) = \{g \in V \mid U^g = U\} for the corresponding normalizer. Let U \leq G be given by its basis (u_1,\dots, u_n). Let N \leq N_G(U) and g \in N. We define \mathrm{Cano}_N^U(g) as the unique reduced preimage of \mathrm{Cano}_{N/U}(gU) under the natural homomorphism N \rightarrow N/U. Now we consider two subgroups W and V of G, and write W_i = W \cap G_i for 1 \leq i \leq n. We define \mathrm{Cano}_V (W ) inductively: suppose that U_{i+1} = \mathrm{Cano}_V (W_i+1) is given by a basis u_{i+1},\dots, u_n together with a conjugating element U_i+1 = W^v_{i+1} and its normalizer N = N_V (U_{i+1}). Suppose that W_i < W_{i+1} and W_i = \langle w_i, W_{i+1}\rangle . Let w_i^\prime be the normalized power of the conjugate w^v_i. Set u_i = \mathrm{Cano}^{U_{i+1}}_N(w^\prime_i), then (u_i,\dots, u_n) is a basis for a subgroup U_i of G_i. We write \mathrm{Cano}_V (W ) for the subgroup U_1 eventually determined by an iterated process. One of the main programs of this package is to compute \mathrm{Cano}_V (W ). This program also computes the normalizer of N_V(W) .
‣ NormalizerNilGroup( G, U ) | ( function ) |
Computes the normalizer of a given subgroup U of G.
‣ IsConjugateSubgroups( G, U, V ) | ( function ) |
Checks if two given subgroups U and V of G are conjugate. If so it returns the conjugating element.
‣ CanonicalConjugateSubgroup( G, U ) | ( function ) |
Computes the canonical conjugate representative of a given subgroup U of G. Returns a record containing the canonical conjugate subgroup, the conjugating element and the normalizer of the given subgroup.
‣ IsCanonicalConjugateSubgroups( G, U, V ) | ( function ) |
Checks if two subgroup U,V of G are conjugate using the canonical representative aproach. If so it returns a record containing the canonical conjugate subgroup, the conjugating elements and the normalizer of the given subgroups.
‣ InfoConjugacySubgroups | ( info class ) |
Info class for the functions of canonical conjugates for subgroups.
‣ CanonicalConjugateList( G, list ) | ( function ) |
Given a list of elements of a nilpotent group G, returns a list of canonical conjugacy representatives of the list and the positions of the elements that belong to the given canonical conjugacy class.
‣ IsConjugateList( G, list ) | ( function ) |
Given a list of elements of a nilpotent group G, returns a list of conjugacy representatives of the list and the positions of the elements that belong to the given conjugacy class.
These algorithms are based on Eddie's work [Lo98].
‣ IntersectionSubgroupsNilGroups( G, U, V ) | ( function ) |
Given two subgroups U,V of a nilpotent group G computes the intersection of both subgroups.
‣ SubgroupProductPair( G, U, V ) | ( function ) |
Given two subgroups U,V of a nilpotent group G, computes the subgroup product pair of both subgroups.
‣ ProductDecomposition( G, U, V, g ) | ( function ) |
Given two subgroups U,V of a nilpotent group G and an element g in G computes the decomposition of g with respect to the product pair of the two subgroups.
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