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<h1>2 Methods and functions</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP002.htm#SECT001">Functions for finite $p$-groups</a>
<li> <A HREF="CHAP002.htm#SECT002">Functions to generate groups and trees</a>
<li> <A HREF="CHAP002.htm#SECT003">Example</a>
</ol>

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


<h2><a name="SECT001">2.1 Functions for finite $p$-groups</a></h2>

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

<a name = "SSEC001.1"></a>
<li><code>LCSFactorTypes( G )</code>

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

<a name = "SSEC001.2"></a>
<li><code>LCSFactorSizes( G )</code>

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

<a name = "SSEC001.3"></a>
<li><code>WidthPGroup( G )</code>

returns the width of G.

<a name = "SSEC001.4"></a>
<li><code>SubgroupRank( G )</code>

returns the (subgroup-)rank of G.

<a name = "SSEC001.5"></a>
<li><code>Obliquity( G )</code>

returns the obliquity of G.

<a name = "SSEC001.6"></a>
<li><code>HasObliquityZero( G )</code>

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


<h2><a name="SECT002">2.2 Functions to generate groups and trees</a></h2>

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.

<a name = "SSEC002.1"></a>
<li><code>GroupsByRankWidthObliquity( p, d, rwo, roots, limit )</code>

determines all p-groups G with G/Φ(G) of order pd 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.

<a name = "SSEC002.2"></a>
<li><code>BranchRWO( G, i, rwo )</code>

for a stable quotient (see <a href="biblio.htm#ER09"><cite>ER09</cite></a>) 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 <code>CHECK_RANK</code> or
<code>CHECK_OBLIQUITY</code> 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 <code>pq</code>-program while using this or the following function,
because such computations will be terminated.

<a name = "SSEC002.3"></a>
<li><code>BoundedDescendantsRWO( G, i, c, rwo)</code>

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

<a name = "SSEC002.4"></a>
<li><code>DrawBranch( branch )</code>

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 <code>gap/xbranch.gi</code>.

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|p2.


<h2><a name="SECT003">2.3 Example</a></h2>

When run under XGap, the following code constructs and draws the
branch with root G/γ5(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(sl2(Q5)).

<pre>
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.
gap> DrawBranch(branch);
</pre>

A window with the following graph should appear.

epsfxsize=8cm epsfboxbranch.ps

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