The p-group generation algorithm by M. F. Newman [1] and E. A. O'Brien [2] [3] is a recursive process for constructing the descendant tree of an assigned finite p-group which is taken as the root of the tree.
is called the exponent-p class (briefly p-class) of
For the convenience of the reader and for pointing out the shifted numeration, we recall that the (usual) lower central series of
A descendant tree is a hierarchical structure for visualizing parent-descendant relations between isomorphism classes of finite p-groups.
The vertices of a descendant tree are isomorphism classes of finite p-groups.
However, a vertex will always be labelled by selecting a representative of the corresponding isomorphism class.
a directed edge of the descendant tree is defined by
The vertices forming the path necessarily coincide with the iterated parents
contains all finite p-groups and is exceptional, since the trivial group
has all the infinitely many elementary abelian p-groups with varying generator rank
However, any non-trivial finite p-group (of order divisible by
Our goal is to compile a complete list of pairwise non-isomorphic immediate descendants of
It turns out that all immediate descendants can be obtained as quotients of a certain extension
For the proof click show on the right hand side.
Therefore, the first part of our goal to compile a list of all immediate descendants of
is an epimorphism and the equivalence classes of allowable subgroups
are precisely the orbits of allowable subgroups under the action of the permutation group
This is precisely what the p-group generation algorithm does in a single step of the recursive procedure for constructing the descendant tree of an assigned root.
is called capable (or extendable) if it possesses at least one immediate descendant, otherwise it is terminal (or a leaf).
For the related phenomenon of multifurcation of a descendant tree at a vertex
The p-group generation algorithm provides the flexibility to restrict the construction of immediate descendants to those of a single fixed step size
, which is very convenient in the case of huge descendant numbers (see the next section).
As concrete examples, we present some interesting finite metabelian p-groups with extensive sets of immediate descendants, using the SmallGroups identifiers and additionally pointing out the numbers
of capable immediate descendants in the usual format
as given by actual implementations of the p-group generation algorithm in the computer algebra systems GAP and MAGMA.
In contrast, groups with abelianization of type
can be viewed as the additive analogue of the multiplicative group
I. R. Shafarevich[4] has proved that the difference between the relation rank
is given by the minimal number of generators of the Schur multiplier of
Furthermore, J. Blackhurst (in the appendix On the nucleus of certain p-groups of a paper by N. Boston, M. R. Bush and F. Hajir [6]) has proved that a non-cyclic finite p-group