As ag groups are groups, all group functions, for example IsAbelian and
Normalizer, can be applied to ag groups. This and the following
sections give further comments on the definition and implementations of
those functions for ag groups. All group functions not mentioned here
are not treated in a special way.
Group( U )
See Group for Ag Groups.
CompositionSeries( G )
Let (g_1, ..., g_n) be an induced generating system of G with respect to the parent group of G. Then for i in {1,...,n} the i.th composition subgroup S_i of the AG system is generated by (g_i, ...,g_n). The n+1.th composition subgroup S_{n+1} is the trivial subgroup of G. The AG series of G is the series {S_1, ..., S_{n+1}}.
Centralizer( U )
The centralizer of an ag group U in its parent group is computed using linear methods while stepping down an elementary abelian series of its parent group.
Centralizer( U, H )
This function call computes the centralizer of H in U using linear methods. H and U must have a common parent.
Centralizer( U, g )
The centralizer of a single element g in an ag group U may be computed whenever g lies in the parent group of U. In that case the same algorithm as for the centralizer of subgroups is used.
ConjugateSubgroup( U, g )
If g is an element of U then U is returned. Otherwise the
remainder of the shifting of g through U is used to conjugate an
induced generating system of U. In that case the information bound to
U.isNilpotent, U.isAbelian, U.isElementaryAbelian and
U.isCyclic, if known, is copied to the conjugate subgroup.
Core( S, U )
AgGroupOps.Core computes successively the core of U stepping up a
composition series of S. See Thi87.
CommutatorSubgroup( G, H )
See CommutatorSubgroup for Ag Groups for details.
ElementaryAbelianSeries( G )
AgGroupOps.ElementaryAbelianSeries returns a series of normal subgroups
of G with elementary abelian factors.
gap> ElementaryAbelianSeries( s4 );
[ s4, Subgroup( s4, [ b, c, d ] ), Subgroup( s4, [ c, d ] ),
Subgroup( s4, [ ] ) ]
gap> d8 := Subgroup( s4, [ a*b^2, c, d ] );
Subgroup( s4, [ a*b^2, c, d ] )
gap> ElementaryAbelianSeries( d8 );
[ Subgroup( s4, [ a*b^2, c, d ] ), Subgroup( s4, [ c, d ] ),
Subgroup( s4, [ ] ) ]
If G is no parent group then AgGroupOps.ElementaryAbelianSeries will
compute a elementary abelian series for the parent group and intersect
this series with G. If G is a parent group then
IsElementaryAbelianAgSeries (see IsElementaryAbelianAgSeries) is used
in order to check if such a series exists. Otherwise an elementary
abelian is computed refining the derived series (see citeLNS84,Gla87).
ElementaryAbelianSeries( L )
L must be a list of ag groups S_1 = H, ..., S_m = {1} with a common parent group such that S_i is a subgroup of S_{i-1} and S_i is normal in G for all i in {2, ..., m}. Then the function returns a series of normal subgroups of G with elementary abelian factors refining the series L.
NormalIntersection( V, W )
If V is an element of the AG series of G, then
AgGroupOps.NormalIntersection uses the depth of V in order to compute
the intersection. Otherwise it uses the Zassenhaus sum-intersection
algorithm (see GS90).
Normalizer( G, U )
SylowSubgroup( G, p )
AgGroupOps.SylowSubgroup uses HallSubgroup (see HallSubgroup) in
order to compute the sylow subgroup of G.
DerivedSeries( G )
AgGroupOps.DerivedSeries uses DerivedSubgroup (see DerivedSubgroup)
in order to compute the derived series of G. It checks if G is normal
in its parent group H. If it is normal all the derived subgroups are
also normal in H. G is always the first element of this list and the
trivial group always the last one since G is soluble.
LowerCentralSeries( G )
AgGroupOps.LowerCentralSeries uses CommutatorSubgroup (see
CommutatorSubgroup) in order to compute the lower central series of
G. It checks if G is normal in its parent group H. If it is
normal all subgroups of the lower central series are also normal in H.
Random( U )
Let (u_1, ..., u_r) be a induced generating system of U. Let e_1, ..., e_r be the relative order of u_1, ..., u_r. Then for r random integers nu_i between 0 and e_i - 1 the word u_1^{nu_1}* ...* u_r^{nu_r} is returned.
IsCyclic( G )
IsFinite( G )
As G is a finite solvable group AgGroupOps.IsFinite returns true.
IsNilpotent( U )
AgGroupOps.IsNilpotent uses Glasby's nilpotency test for ag groups
(see Gla87).
IsNormal( G, U )
IsPerfect( G )
As G is a finite solvable group it is perfect if and only if G is trivial.
IsSubgroup( G, U )
ConjugacyClasses( H )
The conjugacy classes of elements are computed using linear methods. The algorithm depends on the ag series of the parent group of H being a refinement of an elementary abelian series. Thus if this is not true or if H is not a member of the elementary abelian series, an isomorphic group, in which the computation can be done, is created.
The algorithm that is used steps down an elementary abelian series of the parent group of H, basically using affine operation to construct the conjugacy classes of H step by step from its factorgroups.
Orbit( U, pt, op )
AgGroupOps.Orbit returns the orbit of pt under U using the
operation op. The function calls AgOrbitStabilizer in order to
compute the orbit, so please refer to AgOrbitStabilizer for details.
Stabilizer( U, pt, op )
AsGroup( D )
FpGroup( U )
RightCoset( U, g )
AbelianGroup( D, L )
Let L be the list [o_1, ..., o_n] of nonnegative integers o_i > 1.
Then AgWordsOps.AbelianGroup returns the direct product of the cyclic
groups of order o_i using the domain description D. The generators
of these cyclic groups are named beginning with ``a'', ``b'',
``c'', ... followed by a number if o_i is a composite integer.
CyclicGroup( D, n )
See CyclicGroup for Ag Groups.
ElementaryAbelianGroup( D, n )
See ElementaryAbelianGroup for Ag Groups.
DirectProduct( L )
See DirectProduct for Ag Groups.
WreathProduct( G, H, alpha )
See WreathProduct for Ag Groups.
GAP 3.4.4