AgGroupFpGroup( F )
AgGroupFpGroup returns an ag group isomorphic to a finitely presented
finite polycyclic group F.
A finitely presented finite polycyclic group F must be a record with
components generators and relators, such that generators is a list
of abstract generators and relators a list of words in these generators
describing a power-commutator or power-conjugate presentation.
Let g_1, ..., g_n be the generators of F. Then the words of
relators must be the power relators g_k^{e_k} * w_{kk}^{-1} and
commutator relator Comm( g_i, g_j ) * w_{ij}^{-1} or conjugate
relators g_i^{g_j} * w_{ij}^{-1} for all 1 leq k leq n and 1leq
j < i leq n, such that w_{kk} are words in g_{k+1}, ..., g_n and
w_{ij} are words in g_{j+1}, ..., g_n. It is possible to omit some
of the commutator or conjugate relators. Pairs of generators without
commutator or conjugate relator are assumed to commute.
The relative order e_i of g_i need not to be primes, but as all
functions for ag groups need a PAG system, not only an AG system, you
must refine the AG series using RefinedAgSeries (see RefinedAgSeries)
in case some e_i are composite numbers.
Note that it is not checked if the AG presentation is consistent. You
can use IsConsistent (see IsConsistent) to test the consistency of a
presentation. Inconsistent presentations may cause other ag group
functions to return incorrect results.
Initially a collector from the left following the algorithm described in
LS90 is used in order to collect elements of the ag group. This
could be changed using ChangeCollector (see ChangeCollector).
Note that AgGroup will not work with finitely presented groups, you
must use the function AgGroupFpGroup instead. As no checks are done
you can construct an ag group with inconsistent presentation using
AgGroupFpGroup.
GAP 3.4.4