Descendant tree (group theory)

Pro-p groups and coclass trees

For a sound understanding of coclass trees as a particular instance of descendant trees, it is necessary to summarize some facts concerning infinite topological pro-p groups. The members ${\displaystyle {\gamma }_j(S)}$, with ${\displaystyle j\ge 1}$, of the lower central series of a pro-p group ${\displaystyle S}$ are closed (and open) subgroups of finite index, and therefore the corresponding quotients ${\displaystyle S/{\gamma }_j(S)}$ are finite p-groups. The pro-p group ${\displaystyle S}$ is said to be of coclass ${\displaystyle \mathrm{c}\mathrm{c}(S)=r}$ when the limit ${\displaystyle r=\underset{j\to \mathrm{\infty }}{lim}\mathrm{c}\mathrm{c}(S/{\gamma }_j(S))}$ of the coclass of the successive quotients exists and is finite. An infinite pro-p group ${\displaystyle S}$ of coclass ${\displaystyle r}$ is a p-adic pre-space group , since it has a normal subgroup ${\displaystyle T}$, the translation group, which is a free module over the ring ${\displaystyle {\mathbb{Z}}_p}$ of p-adic integers of uniquely determined rank ${\displaystyle d}$, the dimension, such that the quotient ${\displaystyle P=S/T}$ is a finite p-group, the point group, which acts on ${\displaystyle T}$ uniserially. The dimension is given by

${\displaystyle (4)d=(p-1)p^s}$, with some ${\displaystyle 0\le s<r}$.

A central finiteness result for infinite pro-p groups of coclass ${\displaystyle r}$ is provided by the so-called Theorem D, which is one of the five Coclass Theorems proved in 1994 independently by A. Shalev and by C. R. Leedham-Green , and conjectured in 1980 already by C. R. Leedham-Green and M. F. Newman. Theorem D asserts that there are only finitely many isomorphism classes of infinite pro-p groups of coclass ${\displaystyle r}$, for any fixed prime ${\displaystyle p}$ and any fixed non-negative integer ${\displaystyle r}$. As a consequence, if ${\displaystyle S}$ is an infinite pro-p group of coclass ${\displaystyle r}$, then there exists a minimal integer ${\displaystyle i\ge 1}$ such that the following three conditions are satisfied for any integer ${\displaystyle j\ge i}$.

S

1. ${\displaystyle \mathrm{c}\mathrm{c}(S/{\gamma }_j(S))=r}$,
2. ${\displaystyle S/{\gamma }_j(S)}$ is not a lower central quotient of any infinite pro-p group of coclass ${\displaystyle r}$ which is not isomorphic to ${\displaystyle S}$,
3. ${\displaystyle {\gamma }_j/{\gamma }_{j+1}(S)}$ is cyclic of order ${\displaystyle p}$.

The descendant tree ${\displaystyle \mathcal{T}(R)}$, with respect to the parent definition (P2), of the root ${\displaystyle R=S/{\gamma }_i(S)}$ with minimal ${\displaystyle i}$ is called the coclass tree ${\displaystyle \mathcal{T}(S)}$ of ${\displaystyle S}$ and its unique maximal infinite (reverse-directed) path

${\displaystyle (5)R=S/{\gamma }_i(S)\leftarrow S/{\gamma }_{i+1}(S)\leftarrow S/{\gamma }_{i+2}(S)\leftarrow \cdots }$

is called the mainline (or trunk) of the tree.

Tree diagram

Further terminology, used in diagrams visualizing finite parts of descendant trees, is explained in Figure 1 by means of an artificial abstract tree. On the left hand side, a level indicates the basic top-down design of a descendant tree. For concrete trees, such as those in Figure 2, resp. Figure 3, etc., the level is usually replaced by a scale of orders increasing from the top to the bottom. A vertex is capable (or extendable) if it has at least one immediate descendant, otherwise it is terminal (or a leaf). Vertices sharing a common parent are called siblings.

If the descendant tree is a coclass tree ${\displaystyle \mathcal{T}(R)}$ with root ${\displaystyle R=R_0}$ and with mainline vertices ${\displaystyle (R_n{)}_{n\ge 0}}$ labelled according to the level ${\displaystyle n}$, then the finite subtree defined as the difference set

${\displaystyle (6)\mathcal{B}(n)=\mathcal{T}(R_n)\setminus \mathcal{T}(R_{n+1})}$

is called the nth branch (or twig) of the tree or also the branch ${\displaystyle \mathcal{B}(R_n)}$ with root ${\displaystyle R_n}$, for any ${\displaystyle n\ge 0}$. The depth of a branch is the maximal length of the paths connecting its vertices with its root. Figure 1 shows an artificial abstract coclass tree whose branches ${\displaystyle \mathcal{B}(2)}$ and ${\displaystyle \mathcal{B}(4)}$ both have depth ${\displaystyle 0}$, and the branches ${\displaystyle \mathcal{B}(5)\simeq \mathcal{B}(7)}$ and ${\displaystyle \mathcal{B}(6)\simeq \mathcal{B}(8)}$ are pairwise isomorphic as graphs. If all vertices of depth bigger than a given integer ${\displaystyle k\ge 0}$ are removed from the branch ${\displaystyle \mathcal{B}(n)}$, then we obtain the depth-${\displaystyle k}$ pruned branch ${\displaystyle {\mathcal{B}}_k(n)}$. Correspondingly, the depth-${\displaystyle k}$ pruned coclass tree ${\displaystyle {\mathcal{T}}_k(R)}$, resp. the entire coclass tree ${\displaystyle \mathcal{T}(R)}$, consists of the infinite sequence of its pruned branches ${\displaystyle ({\mathcal{B}}_k(n){)}_{n\ge 0}}$, resp. branches ${\displaystyle (\mathcal{B}(n){)}_{n\ge 0}}$, connected by the mainline, whose vertices ${\displaystyle R_n}$ are called infinitely capable.

Virtual periodicity

The periodicity of branches of depth-pruned coclass trees has been proved with analytic methods using zeta functions of groups by M. du Sautoy , and with algebraic techniques using cohomology groups by B. Eick and C. R. Leedham-Green . The former methods admit the qualitative insight of ultimate virtual periodicity, the latter techniques determine the quantitative structure.

Theorem. For any infinite pro-p group ${\displaystyle S}$ of coclass ${\displaystyle r\ge 1}$ and dimension ${\displaystyle d}$, and for any given depth ${\displaystyle k\ge 1}$, there exists an effective minimal lower bound ${\displaystyle f(k)\ge 1}$, where periodicity of length ${\displaystyle d}$ of pruned branches of the coclass tree ${\displaystyle \mathcal{T}(S)}$ sets in, that is, there exist graph isomorphisms

${\displaystyle (7){\mathcal{B}}_k(n+d)\simeq {\mathcal{B}}_k(n)}$ for all ${\displaystyle n\ge f(k)}$.

For the proof, click show on the right hand side.

These central results can be expressed ostensively: When we look at a coclass tree through a pair of blinkers and ignore a finite number of pre-periodic branches at the top, then we shall see a repeating finite pattern (ultimate periodicity). However, if we take wider blinkers the pre-periodic initial section may become longer (virtual periodicity).

The vertex ${\displaystyle P=R_{f(k)}}$ is called the periodic root of the pruned coclass tree, for a fixed value of the depth ${\displaystyle k}$. See Figure 1.

Multifurcation and coclass graphs

Assume that parents of finite p-groups are defined as last non-trivial lower central quotients (P2). For a p-group ${\displaystyle G}$ of coclass ${\displaystyle \mathrm{c}\mathrm{c}(G)=r}$, we can distinguish its (entire) descendant tree ${\displaystyle \mathcal{T}(G)}$ and its coclass-${\displaystyle r}$ descendant tree ${\displaystyle {\mathcal{T}}^r(G)}$, that is the subtree consisting of descendants of coclass ${\displaystyle r}$ only. The group ${\displaystyle G}$ is called coclass-settled if ${\displaystyle \mathcal{T}(G)={\mathcal{T}}^r(G)}$, i.e., if there are no descendants of ${\displaystyle G}$ with bigger coclass than ${\displaystyle r}$.

The nuclear rank ${\displaystyle \nu (G)}$ of ${\displaystyle G}$ in the theory of the p-group generation algorithm by M. F. Newman and E. A. O'Brien provides the following criteria.

N

1. ${\displaystyle G}$ is terminal, and thus trivially coclass-settled, if and only if ${\displaystyle \nu (G)=0}$.
2. If ${\displaystyle \nu (G)=1}$, then ${\displaystyle G}$ is capable, but it remains unknown whether ${\displaystyle G}$ is coclass-settled.
3. If ${\displaystyle \nu (G)=m\ge 2}$, then ${\displaystyle G}$ is capable and definitely not coclass-settled.

In the last case, a more precise assertion is possible: If ${\displaystyle G}$ has coclass ${\displaystyle r}$ and nuclear rank ${\displaystyle \nu (G)=m\ge 2}$, then it gives rise to an m-fold multifurcation into a regular coclass-r descendant tree ${\displaystyle {\mathcal{T}}^r(G)}$ and ${\displaystyle m-1}$ irregular descendant graphs ${\displaystyle {\mathcal{T}}^{r+j}(G)}$ of coclass ${\displaystyle r+j}$, for ${\displaystyle 1\le j\le m-1}$. Consequently, the descendant tree of ${\displaystyle G}$ is the disjoint union

${\displaystyle (8)\mathcal{T}(G)={\dot{\cup }}_{j=0}^{m-1}{\mathcal{T}}^{r+j}(G)}$.

Multifurcation is correlated with different orders of the last non-trivial lower central of immediate descendants. Since the nilpotency class increases exactly by a unit, ${\displaystyle c=\mathrm{c}\mathrm{l}(Q)=\mathrm{c}\mathrm{l}(P)+1}$, from a parent ${\displaystyle P=Q/{\gamma }_c(Q)=\pi (Q)}$ to any immediate descendant ${\displaystyle Q}$, the coclass remains stable, ${\displaystyle r=\mathrm{c}\mathrm{c}(Q)=\mathrm{c}\mathrm{c}(P)}$, if the last non-trivial lower central is cyclic of order ${\displaystyle |{\gamma }_c(Q)|=p}$, since then the exponent of the order also increases exactly by a unit, ${\displaystyle |Q|=p\cdot |P|}$ . In this case, ${\displaystyle Q}$ is a regular immediate descendant with directed edge ${\displaystyle P\leftarrow Q}$ of step size ${\displaystyle 1}$, as usual. However, the coclass increases by ${\displaystyle m-1}$, if ${\displaystyle |{\gamma }_c(Q)|=p^m}$ with ${\displaystyle m\ge 2}$. Then ${\displaystyle Q}$ is called an irregular immediate descendant with directed edge ${\displaystyle P\leftarrow Q}$ of step size ${\displaystyle m}$.

If the condition of step size ${\displaystyle 1}$ is imposed on all directed edges, then the maximal descendant tree ${\displaystyle \mathcal{T}(1)}$ of the trivial group ${\displaystyle 1}$ splits into a countably infinite disjoint union

${\displaystyle (9)\mathcal{T}(1)={\dot{\cup }}_{r=0}^{\mathrm{\infty }}\mathcal{G}(p,r)}$

of directed coclass graphs ${\displaystyle \mathcal{G}(p,r)}$, which are rather forests than trees. More precisely, the above-mentioned Coclass Theorems imply that

${\displaystyle (10)\mathcal{G}(p,r)=\left({\dot{\cup }}_i\mathcal{T}(S_i)\right)\dot{\cup }{\mathcal{G}}_0(p,r)}$

is the disjoint union of finitely many coclass trees ${\displaystyle \mathcal{T}(S_i)}$ of pairwise non-isomorphic infinite pro-p groups ${\displaystyle S_i}$ of coclass ${\displaystyle r}$ (Theorem D) and a finite subgraph ${\displaystyle {\mathcal{G}}_0(p,r)}$ of sporadic groups lying outside of any coclass tree.

Identifiers

The SmallGroups Library identifiers of finite groups, in particular of finite p-groups, given in the form

${\displaystyle ⟨\text{order},\text{counting number}⟩}$

in the following concrete examples of descendant trees, are due to H. U. Besche, B. Eick and E. A. O'Brien . When the group orders are given in a scale on the left hand side, as in Figure 2 and Figure 3, the identifiers are briefly denoted by

${\displaystyle ⟨\text{counting number}⟩}$.

Depending on the prime ${\displaystyle p}$, there is an upper bound on the order of groups for which a SmallGroup identifier exists, e.g. ${\displaystyle 512=2^9}$ for ${\displaystyle p=2}$, and ${\displaystyle 6561=3^8}$ for ${\displaystyle p=3}$. For groups of bigger orders, a notation with generalized identifiers resembling the descendant structure is employed. A regular immediate descendant, connected by an edge of step size ${\displaystyle 1}$ with its parent ${\displaystyle P}$, is denoted by

${\displaystyle P-\mathrm{\#}1;\text{counting number}}$,

and an irregular immediate descendant, connected by an edge of step size ${\displaystyle s\ge 2}$ with its parent ${\displaystyle P}$, is denoted by

${\displaystyle P-\mathrm{\#}s;\text{counting number}}$.

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. A. Ascione in 1979 .

Concrete examples of trees

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.

Coclass 0

The coclass graph

${\displaystyle (11)\mathcal{G}(p,0)={\mathcal{G}}_0(p,0)}$

of finite p-groups of coclass ${\displaystyle 0}$ does not contain any coclass tree and thus exclusively consists of sporadic groups, namely the trivial group ${\displaystyle 1}$ and the cyclic group ${\displaystyle C_p}$ of order ${\displaystyle p}$, which is a leaf (however, it is capable with respect to the lower exponent-p central series). For ${\displaystyle p=2}$ the SmallGroup identifier of ${\displaystyle C_p}$ is ${\displaystyle ⟨2,1⟩}$, for ${\displaystyle p=3}$ it is ${\displaystyle ⟨3,1⟩}$.

Coclass 1

The coclass graph

${\displaystyle (12)\mathcal{G}(p,1)={\mathcal{T}}^1(R)\dot{\cup }{\mathcal{G}}_0(p,1)}$

of finite p-groups of coclass ${\displaystyle 1}$, also called of maximal class, consists of the unique coclass tree ${\displaystyle {\mathcal{T}}^1(R)}$ with root ${\displaystyle R=C_p\times C_p}$, the elementary abelian p-group of rank ${\displaystyle 2}$, and a single isolated vertex (a terminal orphan without proper parent in the same coclass graph, since the directed edge to the trivial group ${\displaystyle 1}$ has step size ${\displaystyle 2}$), the cyclic group ${\displaystyle C_{p^2}}$ of order ${\displaystyle p^2}$ in the sporadic part ${\displaystyle {\mathcal{G}}_0(p,1)}$ (however, this group is capable with respect to the lower exponent-p central series). The tree ${\displaystyle {\mathcal{T}}^1(R)={\mathcal{T}}^1(S_1)}$ is the coclass tree of the unique infinite pro-p group ${\displaystyle S_1}$ of coclass ${\displaystyle 1}$.

For ${\displaystyle p=2}$, resp. ${\displaystyle p=3}$, the SmallGroup identifier of the root ${\displaystyle R}$ is ${\displaystyle ⟨4,2⟩}$, resp. ${\displaystyle ⟨9,2⟩}$, and a tree diagram of the coclass graph from branch ${\displaystyle \mathcal{B}(2)}$ down to branch ${\displaystyle \mathcal{B}(7)}$ (counted with respect to the p-logarithm of the order of the branch root) is drawn in Figure 2, resp. Figure 3, where all groups of order at least ${\displaystyle p^3}$ are metabelian, that is non-abelian with derived length ${\displaystyle 2}$ (vertices represented by black discs in contrast to contour squares indicating abelian groups). In Figure 3, smaller black discs denote metabelian 3-groups where even the maximal subgroups are non-abelian, a feature which does not occur for the metabelian 2-groups in Figure 2, since they all possess an abelian subgroup of index ${\displaystyle p}$ (usually exactly one). The coclass tree of ${\displaystyle \mathcal{G}(2,1)}$, resp. ${\displaystyle \mathcal{G}(3,1)}$, has periodic root ${\displaystyle ⟨8,3⟩}$ and periodicity of length ${\displaystyle 1}$ starting with branch ${\displaystyle \mathcal{B}(3)}$, resp. periodic root ${\displaystyle ⟨81,9⟩}$ and periodicity of length ${\displaystyle 2}$ setting in with branch ${\displaystyle \mathcal{B}(4)}$. Both trees have branches of bounded depth ${\displaystyle 1}$, so their virtual periodicity is in fact a strict periodicity.

However, the coclass tree of ${\displaystyle \mathcal{G}(p,1)}$ with ${\displaystyle p\ge 5}$ has unbounded depth and contains non-metabelian groups, and the coclass tree of ${\displaystyle \mathcal{G}(p,1)}$ with ${\displaystyle p\ge 7}$ has even unbounded width, that is, the number of descendants of a fixed order increases indefinitely with growing order .

With the aid of kernels and targets of Artin transfers, the diagrams in Figure 2 and Figure 3 can be endowed with additional information and redrawn as structured descendant trees.

The concrete examples ${\displaystyle \mathcal{G}(2,1)}$ and ${\displaystyle \mathcal{G}(3,1)}$ of coclass graphs provide an opportunity to give a parametrized polycyclic power-commutator presentation for the complete coclass tree ${\displaystyle {\mathcal{T}}^1(R)\subset \mathcal{G}(p,1)}$, ${\displaystyle p\in \{2,3\}}$, mentioned in the lead section as a benefit of the descendant tree concept and as a consequence of the periodicity of the entire coclass tree. In both cases, a group ${\displaystyle G\in {\mathcal{T}}^1(R)}$ is generated by two elements ${\displaystyle x,y}$ but the presentation contains the series of higher commutators ${\displaystyle s_j=[s_{j-1},x]}$, ${\displaystyle 3\le j\le n-1=\mathrm{c}\mathrm{l}(G)}$, starting with the main commutator ${\displaystyle s_2=[y,x]}$. The nilpotency is formally expressed by the relation ${\displaystyle s_n=1}$, when the group is of order ${\displaystyle |G|=p^n}$.

For ${\displaystyle p=2}$, there are two parameters ${\displaystyle 0\le w,z\le 1}$ and the pc-presentation is given by

(13) ${\displaystyle \begin{array}{cc}{G}^n(z,w)=& ⟨x,y,s_2,\dots ,s_{n-1}\mid \\ & x^2=s_{n-1}^w,y^2=s_2^{-1}{s}_{n-1}^z,[s_2,y]=1,\\ & s_2=[y,x],s_j=[s_{j-1},x]\text{ for }3\le j\le n-1⟩\end{array}}$

The 2-groups of maximal class, that is of coclass ${\displaystyle 1}$, form three periodic infinite sequences,

• the dihedral groups, ${\displaystyle D(2^n)=G^n(0,0)}$, ${\displaystyle n\ge 3}$, forming the mainline (with infinitely capable vertices),
• the generalized quaternion groups, ${\displaystyle Q(2^n)=G^n(0,1)}$, ${\displaystyle n\ge 3}$, which are all terminal vertices,
• the semidihedral groups, ${\displaystyle S(2^n)=G^n(1,0)}$, ${\displaystyle n\ge 4}$, which are also leaves.

For ${\displaystyle p=3}$, there are three parameters ${\displaystyle 0\le a\le 1}$ and ${\displaystyle -1\le w,z\le 1}$ and the pc-presentation is given by

(14) ${\displaystyle \begin{array}{cc}{G}_a^n(z,w)=& ⟨x,y,s_2,\dots ,s_{n-1}\mid \\ & x^3=s_{n-1}^w,y^3=s_2^{-3}{s}_3^{-1}{s}_{n-1}^z,[y,s_2]=s_{n-1}^a,\\ & s_2=[y,x],s_j=[s_{j-1},x]\text{ for }3\le j\le n-1⟩\end{array}}$

3-groups with parameter ${\displaystyle a=0}$ possess an abelian maximal subgroup, those with parameter ${\displaystyle a=1}$ do not. More precisely, an existing abelian maximal subgroup is unique, except for the two extra special groups ${\displaystyle G_0^3(0,0)}$ and ${\displaystyle G_0^3(0,1)}$, where all four maximal subgroups are abelian.

In contrast to any bigger coclass ${\displaystyle r\ge 2}$, the coclass graph ${\displaystyle \mathcal{G}(p,1)}$ exclusively contains p-groups ${\displaystyle G}$ with abelianization ${\displaystyle G/G^{\mathrm{\prime }}}$ of type ${\displaystyle (p,p)}$, except for its unique isolated vertex ${\displaystyle C_{p^2}}$. The case ${\displaystyle p=2}$ is distinguished by the truth of the reverse statement: Any 2-group with abelianization of type ${\displaystyle (2,2)}$ is of coclass ${\displaystyle 1}$ (O. Taussky's Theorem ).

Coclass 2

The genesis of the coclass graph ${\displaystyle \mathcal{G}(p,r)}$ with ${\displaystyle r\ge 2}$ is not uniform. p-groups with several distinct abelianizations contribute to its constitution. For coclass ${\displaystyle r=2}$, there are essential contributions from groups ${\displaystyle G}$ with abelianizations ${\displaystyle G/G^{\mathrm{\prime }}}$ of the types ${\displaystyle (p,p)}$, ${\displaystyle (p^2,p)}$, ${\displaystyle (p,p,p)}$, and an isolated contribution by the cyclic group ${\displaystyle C_{p^3}}$ of order ${\displaystyle p^3}$:

${\displaystyle (15)\mathcal{G}(p,2)={\mathcal{G}}_{(p,p)}(p,2)\dot{\cup }{\mathcal{G}}_{(p^2,p)}(p,2)\dot{\cup }{\mathcal{G}}_{(p,p,p)}(p,2)\dot{\cup }{\mathcal{G}}_{(p^3)}(p,2)}$.

Abelianization of type (p,p)

As opposed to p-groups of coclass ${\displaystyle 2}$ with abelianization of type ${\displaystyle (p^2,p)}$ or ${\displaystyle (p,p,p)}$, which arise as regular descendants of abelian p-groups of the same types, p-groups of coclass ${\displaystyle 2}$ with abelianization of type ${\displaystyle (p,p)}$ arise from irregular descendants of a non-abelian p-group of coclass ${\displaystyle 1}$ which is not coclass-settled.

For the prime ${\displaystyle p=2}$, such groups do not exist at all, since the 2-group ${\displaystyle ⟨8,3⟩}$ is coclass settled, which is the deeper reason for Taussky's Theorem. This remarkable fact has been observed by Giuseppe Bagnera in 1898 already.

For odd primes ${\displaystyle p\ge 3}$, the existence of p-groups of coclass ${\displaystyle 2}$ with abelianization of type ${\displaystyle (p,p)}$ is due to the fact that the group ${\displaystyle G_0^3(0,0)}$ is not coclass-settled. Its nuclear rank equals ${\displaystyle 2}$, which gives rise to a bifurcation of the descendant tree ${\displaystyle \mathcal{T}(G_0^3(0,0))}$ into two coclass graphs. The regular component ${\displaystyle {\mathcal{T}}^1(G_0^3(0,0))}$ is a subtree of the unique tree ${\displaystyle {\mathcal{T}}^1(C_p\times C_p)}$ in the coclass graph ${\displaystyle \mathcal{G}(p,1)}$. The irregular component ${\displaystyle {\mathcal{T}}^2(G_0^3(0,0))}$ becomes a subgraph ${\displaystyle \mathcal{G}={\mathcal{G}}_{(p,p)}(p,2)}$ of the coclass graph ${\displaystyle \mathcal{G}(p,2)}$ when the connecting edges of step size ${\displaystyle 2}$ of the irregular immediate descendants of ${\displaystyle G_0^3(0,0)}$ are removed.

For ${\displaystyle p=3}$, this subgraph ${\displaystyle \mathcal{G}}$ is drawn in Figure 4, which shows the interface between finite 3-groups with coclass ${\displaystyle 1}$ and ${\displaystyle 2}$ of type ${\displaystyle (3,3)}$. ${\displaystyle \mathcal{G}}$ has seven top level vertices of three important kinds, all having order ${\displaystyle 243=3^5}$, which have been discovered by G. Bagnera .

• Firstly, there are two terminal Schur σ-groups ${\displaystyle ⟨243,5⟩}$ and ${\displaystyle ⟨243,7⟩}$ in the sporadic part ${\displaystyle {\mathcal{G}}_0(3,2)}$ of the coclass graph ${\displaystyle \mathcal{G}(3,2)}$.
• Secondly, the two groups ${\displaystyle G=⟨243,4⟩}$ and ${\displaystyle G=⟨243,9⟩}$ are roots of finite trees ${\displaystyle {\mathcal{T}}^2(G)}$ in the sporadic part ${\displaystyle {\mathcal{G}}_0(3,2)}$. However, since they are not coclass-settled, the complete trees ${\displaystyle \mathcal{T}(G)}$ are infinite .
• Finally, the three groups ${\displaystyle ⟨243,3⟩}$, ${\displaystyle ⟨243,6⟩}$ and ${\displaystyle ⟨243,8⟩}$ give rise to (infinite) coclass trees, e.g., ${\displaystyle {\mathcal{T}}^2(⟨729,40⟩)}$, ${\displaystyle {\mathcal{T}}^2(⟨243,6⟩)}$, ${\displaystyle {\mathcal{T}}^2(⟨243,8⟩)}$, each having a metabelian mainline, in the coclass graph ${\displaystyle \mathcal{G}(3,2)}$. None of these three groups is coclass-settled.

Displaying additional information on kernels and targets of Artin transfers, we can draw these trees as structured descendant trees.

Definition. Generally, a Schur group (called a closed group by I. Schur, who coined the concept) is a pro-p group ${\displaystyle G}$ whose relation rank ${\displaystyle d_2(G)={\mathrm{d}\mathrm{i}\mathrm{m}}_{{\mathbb{F}}_p}({\mathrm{H}}^2(G,{\mathbb{F}}_p))}$ coincides with its generator rank ${\displaystyle d_1(G)={\mathrm{d}\mathrm{i}\mathrm{m}}_{{\mathbb{F}}_p}({\mathrm{H}}^1(G,{\mathbb{F}}_p))}$. A σ-group is a pro-p group ${\displaystyle G}$ which possesses an automorphism ${\displaystyle \sigma \in \mathrm{A}\mathrm{u}\mathrm{t}(G)}$ inducing the inversion ${\displaystyle x\mapsto x^{-1}}$ on its abelianization ${\displaystyle G/G^{\mathrm{\prime }}}$. A Schur σ-group is a Schur group ${\displaystyle G}$ which is also a σ-group and has a finite abelianization ${\displaystyle G/G^{\mathrm{\prime }}}$.

${\displaystyle ⟨243,3⟩}$ is not root of a coclass tree,

since its immediate descendant ${\displaystyle ⟨729,40⟩}$, which is root of a coclass tree with metabelian mainline vertices, has two siblings ${\displaystyle ⟨729,35⟩}$, resp. ${\displaystyle ⟨729,34⟩}$, which give rise to a single, resp. three, coclass tree(s) with non-metabelian mainline vertices having cyclic centres of order ${\displaystyle 3}$ and branches of considerable complexity but nevertheless of bounded depth ${\displaystyle 5}$.

Pro-3 groups of coclass 2 with non-trivial centre

B. Eick, C. R. Leedham-Green, M. F. Newman and E. A. O'Brien have constructed a family of infinite pro-3 groups with coclass ${\displaystyle 2}$ having a non-trivial centre of order ${\displaystyle 3}$. The family members are characterized by three parameters ${\displaystyle (f,g,h)}$. Their finite quotients generate all mainline vertices with bicyclic centres of type ${\displaystyle (3,3)}$ of six coclass trees in the coclass graph ${\displaystyle \mathcal{G}(3,2)}$. The association of parameters to the roots of these six trees is given in Table 1, the tree diagrams, except for the abelianization ${\displaystyle (3,3,3)}$, are indicated in Figure 4 and Figure 5, and the parametrized pro-3 presentation is given by

(16) ${\displaystyle \begin{array}{cc}G(f,g,h)=& ⟨a,t,z\mid \\ & a^3=z^f,[t,t^a]=z^g,t^{1+a+a^2}=z^h,\\ & z^3=1,[z,a]=1,[z,t]=1⟩\end{array}}$

Abelianization of type (p²,p)

For ${\displaystyle p=3}$, the top levels of the subtree ${\displaystyle {\mathcal{T}}^2(⟨27,2⟩)}$ of the coclass graph ${\displaystyle \mathcal{G}(3,2)}$ are drawn in Figure 5. The most important vertices of this tree are the eight siblings sharing the common parent ${\displaystyle ⟨81,3⟩}$, which are of three important kinds.

• Firstly, there are three leaves ${\displaystyle ⟨243,20⟩}$, ${\displaystyle ⟨243,19⟩}$, ${\displaystyle ⟨243,16⟩}$ having cyclic centre of order ${\displaystyle 9}$, and a single leaf ${\displaystyle ⟨243,18⟩}$ with bicyclic centre of type ${\displaystyle (3,3)}$.
• Secondly, the group ${\displaystyle G=⟨243,14⟩}$ is root of a finite tree ${\displaystyle \mathcal{T}(G)={\mathcal{T}}^2(G)}$.
• Finally, the three groups ${\displaystyle ⟨243,13⟩}$, ${\displaystyle ⟨243,15⟩}$ and ${\displaystyle ⟨243,17⟩}$ give rise to infinite coclass trees, e.g., ${\displaystyle {\mathcal{T}}^2(⟨2187,319⟩)}$, ${\displaystyle {\mathcal{T}}^2(⟨243,15⟩)}$, ${\displaystyle {\mathcal{T}}^2(⟨243,17⟩)}$, each having a metabelian mainline, the first with cyclic centres of order ${\displaystyle 3}$, the second and third with bicyclic centres of type ${\displaystyle (3,3)}$.

Here, ${\displaystyle ⟨243,13⟩}$ is not root of a coclass tree, since aside from its descendant ${\displaystyle ⟨2187,319⟩}$, which is root of a coclass tree with metabelian mainline vertices, it possesses five further descendants which give rise to coclass trees with non-metabelian mainline vertices having cyclic centres of order ${\displaystyle 3}$ and branches of extreme complexity, here partially even with unbounded depth.

Abelianization of type (p,p,p)

For ${\displaystyle p=2}$, resp. ${\displaystyle p=3}$, there exists a unique coclass tree with p-groups of type ${\displaystyle (p,p,p)}$ in the coclass graph ${\displaystyle \mathcal{G}(p,2)}$. Its root is the elementary abelian p-group of type ${\displaystyle (p,p,p)}$, that is, ${\displaystyle ⟨8,5⟩}$, resp. ${\displaystyle ⟨27,5⟩}$. This unique tree corresponds to the pro-2 group of the family ${\displaystyle \mathrm{\#}59}$ by M. F. Newman and E. A. O'Brien, resp. to the pro-3 group given by the parameters ${\displaystyle (f,g,h)=(0,0,0)}$ in Table 1. For ${\displaystyle p=2}$, the tree is indicated in Figure 6, which shows some finite 2-groups with coclass ${\displaystyle 2,3,4}$ of type ${\displaystyle (2,2,2)}$.

Coclass 3

Here again, p-groups with several distinct abelianizations contribute to the constitution of the coclass graph ${\displaystyle \mathcal{G}(p,3)}$. There are regular, resp. irregular, essential contributions from groups ${\displaystyle G}$ with abelianizations ${\displaystyle G/G^{\mathrm{\prime }}}$ of the types ${\displaystyle (p^3,p)}$, ${\displaystyle (p^2,p^2)}$, ${\displaystyle (p^2,p,p)}$, ${\displaystyle (p,p,p,p)}$, resp. ${\displaystyle (p,p)}$, ${\displaystyle (p^2,p)}$, ${\displaystyle (p,p,p)}$, and an isolated contribution by the cyclic group ${\displaystyle C_{p^4}}$ of order ${\displaystyle p^4}$.

Abelianization of type (p,p,p)

Since the elementary abelian p-group ${\displaystyle C_p\times C_p\times C_p}$ of rank ${\displaystyle 3}$, that is, ${\displaystyle ⟨8,5⟩}$, resp. ${\displaystyle ⟨27,5⟩}$, for ${\displaystyle p=2}$, resp. ${\displaystyle p=3}$, is not coclass-settled, it gives rise to a multifurcation. The regular component ${\displaystyle {\mathcal{T}}^2(C_p\times C_p\times C_p)}$ has been described in the section about coclass ${\displaystyle 2}$. The irregular component ${\displaystyle {\mathcal{T}}^3(C_p\times C_p\times C_p)}$ becomes a subgraph ${\displaystyle \mathcal{G}={\mathcal{G}}_{(p,p,p)}(p,3)}$ of the coclass graph ${\displaystyle \mathcal{G}(p,3)}$ when the connecting edges of step size ${\displaystyle 2}$ of the irregular immediate descendants of ${\displaystyle C_p\times C_p\times C_p}$ are removed.

For ${\displaystyle p=2}$, this subgraph ${\displaystyle \mathcal{G}}$ is contained in Figure 6. It has nine top level vertices of order ${\displaystyle 32=2^5}$ which can be divided into terminal and capable vertices.

• The two groups ${\displaystyle ⟨32,32⟩}$ and ${\displaystyle ⟨32,33⟩}$ are leaves.
• The five groups ${\displaystyle ⟨32,\mathrm{27..31}⟩}$ and the two groups ${\displaystyle ⟨32,\mathrm{34..35}⟩}$ are infinitely capable.

The trees arising from the capable vertices are associated with infinite pro-2 groups by M. F. Newman and E. A. O'Brien in the following manner.

${\displaystyle ⟨32,28⟩}$ gives rise to two trees,

${\displaystyle {\mathcal{T}}^3(⟨64,140⟩)}$ associated with family ${\displaystyle \mathrm{\#}73}$, and

${\displaystyle {\mathcal{T}}^3(⟨64,147⟩)}$ associated with family ${\displaystyle \mathrm{\#}74}$.

${\displaystyle {\mathcal{T}}^3(⟨32,29⟩)}$ is associated with family ${\displaystyle \mathrm{\#}75}$.

${\displaystyle {\mathcal{T}}^3(⟨32,30⟩)}$ is associated with family ${\displaystyle \mathrm{\#}76}$.

${\displaystyle {\mathcal{T}}^3(⟨32,31⟩)}$ is associated with family ${\displaystyle \mathrm{\#}77}$.

${\displaystyle ⟨32,34⟩}$ gives rise to

${\displaystyle {\mathcal{T}}^3(⟨64,174⟩)}$ associated with family ${\displaystyle \mathrm{\#}78}$. Finally,

${\displaystyle {\mathcal{T}}^3(⟨32,35⟩)}$ is associated with family ${\displaystyle \mathrm{\#}79}$.

Hall-Senior classification of 2-groups

Seven of these nine top level vertices have been investigated by E. Benjamin, F. Lemmermeyer and C. Snyder with respect to their occurrence as class-2 quotients ${\displaystyle Q=G/{\gamma }_3(G)}$ of bigger metabelian 2-groups ${\displaystyle G}$ of type ${\displaystyle (2,2,2)}$ and with coclass ${\displaystyle 3}$, which are exactly the members of the descendant trees of the seven vertices. These authors use the classification of 2-groups by M. Hall and J. K. Senior which is put in correspondence with the SmallGroups Library 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 ${\displaystyle 2}$ in ${\displaystyle G}$ are denoted by ${\displaystyle H_i}$, for ${\displaystyle 1\le i\le 7}$.

History

Descendant trees with central quotients as parents (P1) are implicit in P. Hall's 1940 paper 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 . The first extensive tree diagrams have been drawn manually by J. A. Ascione, G. Havas and C. R. Leedham-Green (1977) , by J. A. Ascione (1979) , and by B. Nebelung (1989) . 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).