Descendant tree (group theory)
Consequently, descendant trees play a fundamental role in the classification of finite p-groups.
By means of kernels and targets of Artin transfer homomorphisms, descendant trees can be endowed with additional structure.
The parent definitions (P2–P3) have the advantage that any non-trivial finite p-group (of order divisible by
[2] Theorem D asserts that there are only finitely many isomorphism classes of infinite pro-p groups of coclass
On the left hand side, a level indicates the basic top-down design of a descendant tree.
Figure 3, etc., the level is usually replaced by a scale of orders increasing from the top to the bottom.
The depth of a branch is the maximal length of the paths connecting its vertices with its root.
[8] The former methods admit the qualitative insight of ultimate virtual periodicity, the latter techniques determine the quantitative structure.
is called the periodic root of the pruned coclass tree, for a fixed value of the depth
Assume that parents of finite p-groups are defined as last non-trivial lower central quotients (P2).
, then it gives rise to an m-fold multifurcation into a regular coclass-r descendant tree
Multifurcation is correlated with different orders of the last non-trivial lower central of immediate descendants.
in the following concrete examples of descendant trees, are due to H. U. Besche, B. Eick and E. A. O'Brien .
For groups of bigger orders, a notation with generalized identifiers resembling the descendant structure is employed.
The implementations of the p-group generation algorithm in the computational algebra systems GAP and Magma use these generalized identifiers, which go back to J.
[15] In all examples, the underlying parent definition (P2) corresponds to the usual lower central series.
Occasional differences to the parent definition (P3) with respect to the lower exponent-p central series are pointed out.
, and a single isolated vertex (a terminal orphan without proper parent in the same coclass graph, since the directed edge to the trivial group
(counted with respect to the p-logarithm of the order of the branch root) is drawn in Figure 2, resp.
(vertices represented by black discs in contrast to contour squares indicating abelian groups).
More precisely, an existing abelian maximal subgroup is unique, except for the two extra special groups
three, coclass tree(s) with non-metabelian mainline vertices having cyclic centres of order
B. Eick, C. R. Leedham-Green, M. F. Newman and E. A. O'Brien [5] have constructed a family of infinite pro-3 groups with coclass
The trees arising from the capable vertices are associated with infinite pro-2 groups by M. F. Newman and E. A. O'Brien [6] in the following manner.
Seven of these nine top level vertices have been investigated by E. Benjamin, F. Lemmermeyer and C. Snyder [20] with respect to their occurrence as class-2 quotients
These authors use the classification of 2-groups by M. Hall and J. K. Senior [21] which is put in correspondence with the SmallGroups Library [13] in Table 2.
The complexity of the descendant trees of these seven vertices increases with the 2-ranks and 4-ranks indicated in Table 2, where the maximal subgroups of index
Descendant trees with central quotients as parents (P1) are implicit in P. Hall's 1940 paper [22] about isoclinism of groups.
Trees with last non-trivial lower central quotients as parents (P2) were first presented by C. R. Leedham-Green at the International Congress of Mathematicians in Vancouver, 1974 .
[24] In the former two cases, the parent definition by means of the lower exponent-p central series (P3) was adopted in view of computational advantages, in the latter case, where theoretical aspects were focussed, the parents were taken with respect to the usual lower central series (P2).
