Elements of pc-groups are words. A word is defined inductively as follows:
Given the pc-group G and a sequence Q of length n, containing the distinct positive integers (alpha_i), 0 <= (alpha_i) < p_(i) for i = 1, ..., n, construct the element x of G given by x = a_1^(alpha_1) ... a_n^(alpha_n), 0 <= alpha_i < p_i for i = 1, ..., n.
Given an element x belonging to the pc-group G, where x = a_1^(alpha_1) ... a_n^(alpha_n), 0 <= alpha_i < p_i for i = 1, ..., n, return the sequence Q of n integers defined by Q[i] = (alpha_i), for i = 1, ..., n.
Construct the identity element of the pc-group G.
Throughout this subsection, G will be a pc-group with pc-generators
a_1, ..., a_n.
g * h : GrpPCElt, GrpPCElt -> GrpPCElt
Product of the element g and the element h, where g and h belong to some common subgroup G of a pc-group U. If g and h are given as elements belonging to the same proper subgroup G of U, then the result will be returned as an element of G; if g and h are given as elements belonging to distinct subgroups H and K of U, then the product is returned as an element of G, where G is the smallest subgroup of U known to contain both elements.
Replace g with the product of element g and element h.
The n-th power of the element g, where n is a positive or negative integer.
Replace g with the n-th power of the element g.
Quotient of the element g by the element h. Here g and h must belong to some common subgroup G of a pc-group U. The rules for determining the parent group of g/h are the same as for g * h.
Replace g with the quotient of the element g by the element h.
Conjugate of the element g by the element h, i.e. the element h^(-1) * g * h. Here g and h must belong to some common subgroup G of a pc-group U. The rules for determining the parent group of g^h are the same as for g * h.
Replace g with the conjugate of the element g by the element h.
Given the n words g_1, ..., g_n belonging to some common subgroup G of a pc-group U. If g_1, ..., g_n are given as elements belonging to the same proper subgroup G of U, then the result will be returned as an element of G; if g_1, ..., g_n are given as elements belonging to distinct subgroups of U, then the product is returned as an element of G, where G is the smallest subgroup of U known to contain all elements. Commutators are left-normed, so that they are evaluated from left to right.
Order of the element x.
The parent group G of the element x.
Given elements g and h belonging to a common pc-group, return true if g and h are the same element, false otherwise.
True if g is the identity element, false otherwise.
Given elements g and h belonging to a common pc-group, return true if g and h are distinct elements, false otherwise.
Check: BoolElt Default: true
ExponentLimit: RngIntElt Default: 20
Given a free group F of rank n with generating set X, and a collection R of pc-relations on X, construct the soluble group G defined by the power-conjugate presentation < X | R >.The construct R denotes a list of pc-relations. Thus, an element of R must be one of:
Note the following points:
- A power relation a_j^(p_j) = w_(jj), 1 <= j <= n, where w_(jj) is 1 or a word in generators a_(j + 1), ..., a_n for j < n, and w_(jj) = 1 for j = n, and p_j a prime.
- A conjugate relation a_j^(a_i) = w_(ij), 1 <= i < j <= n, where w_(ij) is a word in the generators a_(i + 1), ..., a_n.
- A power a_j^(p_j), 1 <= j <= n and p_j a prime, which is treated as the power relation a_j^(p_j) = Id(F).
- A conjugate a_j^(a_i), 1 <= i < j <= n, which is treated as the conjugate relation a_j^(a_i) = Id(F).
- A set of (a) -- (d).
- A sequence of (a) -- (d).
In addition, one can alternatively specify a power-commutator presentation using commutators rather than conjugates. However, commutators and conjugates cannot be mixed in a single presentation.
- A power relation must be present for each generator a_i, i = 1, ..., n;
- Trivial conjugates may be omitted;
- The words w_((ij)) must be in normal form.
This constructor returns a pc-group because the category GrpPC is stated. If no category were stated, it would return an fp-group.
The parameters Check and ExponentLimit may be used. Check indicates whether or not the presentation is checked for consistency. ExponentLimit determines the amount of space that will be used by the group to speed calculations. Given ExponentLimit := e, the group will store the products a^i * b^j where a and b are generators and i and j are in the range 1 to e.
The natural homomorphism from F -> G is returned as well.
Check: BoolElt Default: true
ExponentLimit: RngIntElt Default: 20
Construct the soluble group G defined by the power-conjugate presentation < x_1, ..., x_n | R >.The construct x_1, ..., x_n defines names for the generators of G that are local to the constructor, i.e. they are used when writing down the relations to the right of the bar. However, no assignment of values to these variables is made. If the user wants to refer to the generators by these (or other) names, then the generators assignment construct must be used on the left hand side of an assignment statement.
The construct R denotes a list of pc-relations. The syntax and semantics for the relations clause is identical to that appearing in the quo-construct above.
A map f from the free group on x_1, ..., x_n to G is returned as well.
The parameters Check and ExponentLimit may be used as described in the quo-construct above.
\ < a, b, c, d | a^2 = c, b^2, c^2 = e, d^5, e^2, (b,a) = e, (c,a), (c,b), \ (d,a) = d, (d,b), (d,c) = d^3, (e,a), (e,b), (e,c), (e,d) >.Giving the relations in the form of a list, this presentation would be specified as follows:
> G<a,b,c,d,e> := PolycyclicGroup<a, b, c, d, e | > a^2 = c, b^2, c^2 = e, d^5, e^2, > b^a = b*e, d^a = d^2, d^c = d^4 >;Starting from a free group and giving the relations in the form of a set, this presentation would be specified as follows:
> F<a,b,c,d,e> := FreeGroup(5);
> rels := { a^2 = c, b^2 = Id(F), c^2 = e, d^5 = Id(F), e^2 = Id(F),
> b^a = b*e, d^a = d^2, d^c = d^4 };
> G<a,b,c,d,e> := quo< GrpPC : F | rels >;
> G;