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# 2 Methods and functions

### Sections

1. Functions for finite $p$-groups
2. Functions to generate groups and trees
3. Example

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

## 2.1 Functions for finite $p$-groups

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.

  ## 2.2 Functions to generate groups and trees

  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 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.

- `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/gi(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|p2.

  ## 2.3 Example

  When run under XGap, the following code constructs and draws the branch with root G/g5(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)).

      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

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