This chapter describes all the main methods and functions of this package.

Let *G* be a finite *p*-group given by a consistent polycyclic
presentation as Pc group.

`LCSFactorTypes( G )`

returns the abelian invariants of the lower central series factors of *G*.

`LCSFactorSizes( G )`

returns the orders of the lower central series factors of *G*.

`WidthPGroup( G )`

returns the width of *G*.

`SubgroupRank( G )`

returns the (subgroup-)rank of *G*.

`Obliquity( G )`

returns the obliquity of *G*.

`HasObliquityZero( G )`

checks whether *G* has obliquity 0 and returns true or false.

Let *G*(*p*,*rwo*) denote the full tree of all finite *p*-groups with
rank *rwo*[1], width *rwo*[2] and obliquity *rwo*[3]. This tree can be
finite or infinite; if it is infinite, then the infinite
pro-*p*-groups of the considered rank, width and obliquity specify
infinite subtrees of the full tree. The groups not contained in such
an infinite subtree are called sporadic.

`GroupsByRankWidthObliquity( p, d, rwo, roots, limit )`

determines all *p*-groups *G* with *G*/F(*G*) of order *p*^{d} and
rank, width and obliquity as prescribed in *rwo* up to order *limit*.
Here *p* and *d* are integers, *rwo* is a list of three integers and
*limit* is an integer.

The parameter *roots* is a list of groups described by their id's
with respect to the small groups library. The descendants of the
groups described in *roots* are excluded from the output of this
function. This option can be used to prune the tree of groups
determined by this function.

If there are only finitely many sporadic *p*-groups with given rank,
width and obliquity, then this function can be used to generate them;
in this case *roots* must contain a complete list of all id's of roots
of infinite subtrees and *limit* can be set to infinity.

`BranchRWO( G, i, rwo )`

for a stable quotient (see ER09) *G* of a pro-*p*-group of
rank *rwo*[1], *rwo*[2] and obliquity *rwo*[3], this function returns
the *i*-th branch of its corresponding tree. The structure of the tree
is encoded in a list. If one of the global parameters `CHECK_RANK`

or
`CHECK_OBLIQUITY`

is set to false, then checking the corresponding
invariant is omitted and hence a potentially larger tree is returned.

The user is advised not to perform any other computations using ANUPQ
or the `pq`

-program while using this or the following function,
because such computations will be terminated.

`BoundedDescendantsRWO( G, i, c, rwo)`

returns the tree of all descendants of *G*/g_{i}(*G*) of rank *rwo*[1],
width *rwo*[2], obliquity *rwo*[3] and class at most *c*.

`DrawBranch( branch )`

if the package is run under XGap, then this function can be used to draw
a branch as output by the above two functions in the case of width 2.
The user may wish to improve the quality of the output by modifying
the file `gap/xbranch.gi`

.

Vertices drawn on the same level correspond to groups of the same
class. If *G* is a descendant of *H* in the branch, then *G* is drawn
as a filled circle if |*G*| = |*H*|*p* and as a solid
box if |*G*| = |*H*|*p*^{2}.

When run under XGap, the following code constructs and draws the
branch with root *G*/g_{5}(*G*) in the graph of finite 5-groups of
rank 3, width 2 and obliquity 0, where *G* is the Sylow
pro-*p*-subgroup of Aut(*sl*_{2}(**Q**_{5})).

gap> g := ProPSylowGroupOfPSL(2,5,6); Pcp-group with orders [ 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5 ] gap> branch := BranchRWO(g,5,[3,2,0]);; ConstructBranch: root-p-class: 4 Constructed 3 1-step descendants. ConstructBranch: root-p-class: 5 Constructed 0 1-step descendants. Constructed 0 2-step descendants. ConstructBranch: root-p-class: 5 Constructed 0 1-step descendants. Constructed 0 2-step descendants. ConstructBranch: root-p-class: 5 Constructed 0 1-step descendants. Constructed 0 2-step descendants. Constructed 3 2-step descendants. ConstructBranch: root-p-class: 5 Constructed 0 1-step descendants. Constructed 0 2-step descendants. ConstructBranch: root-p-class: 5 Constructed 0 1-step descendants. Constructed 0 2-step descendants. time: 0:00:16.525 gap> DrawBranch(branch);

A window with the following graph should appear.

epsfxsize=8cm epsfboxbranch.ps

fwtree manual

April 2009