According to M. F. Newman,^[1] there exist several distinct definitions of the parent ${\displaystyle \pi (G)}$  of a finite p-group ${\displaystyle G}$ . The common principle is to form the quotient ${\displaystyle \pi (G)=G/N}$  of ${\displaystyle G}$  by a suitable normal subgroup ${\displaystyle N\triangleleft G}$  which can be either

P

1. the centre ${\displaystyle N=\zeta _{1}(G)}$  of ${\displaystyle G}$ , whence ${\displaystyle \pi (G)=G/\zeta _{1}(G)}$  is called the central quotient of ${\displaystyle G}$ , or
2. the last non-trivial term ${\displaystyle N=\gamma _{c}(G)}$  of the lower central series of ${\displaystyle G}$ , where ${\displaystyle c}$  denotes the nilpotency class of ${\displaystyle G}$ , or
3. the last non-trivial term ${\displaystyle N=P_{c-1}(G)}$  of the lower exponent-p central series of ${\displaystyle G}$ , where ${\displaystyle c}$  denotes the exponent-p class of ${\displaystyle G}$ , or
4. the last non-trivial term ${\displaystyle N=G^{(d-1)}}$  of the derived series of ${\displaystyle G}$ , where ${\displaystyle d}$  denotes the derived length of ${\displaystyle G}$ .

In each case, ${\displaystyle G}$  is called an immediate descendant of ${\displaystyle \pi (G)}$  and a directed edge of the tree is defined either by ${\displaystyle G\to \pi (G)}$  in the direction of the canonical projection ${\displaystyle \pi :G\to \pi (G)}$  onto the quotient ${\displaystyle \pi (G)=G/N}$  or by ${\displaystyle \pi (G)\to G}$  in the opposite direction, which is more usual for descendant trees. The former convention is adopted by C. R. Leedham-Green and M. F. Newman,^[2] by M. du Sautoy and D. Segal,^[3] by C. R. Leedham-Green and S. McKay,^[4] and by B. Eick, C. R. Leedham-Green, M. F. Newman and E. A. O'Brien.^[5] The latter definition is used by M. F. Newman,^[1] by M. F. Newman and E. A. O'Brien,^[6] by M. du Sautoy,^[7] and by B. Eick and C. R. Leedham-Green.^[8]

In the following, the direction of the canonical projections is selected for all edges. Then, more generally, a vertex ${\displaystyle R}$  is a descendant of a vertex ${\displaystyle P}$ , and ${\displaystyle P}$  is an ancestor of ${\displaystyle R}$ , if either ${\displaystyle R}$  is equal to ${\displaystyle P}$  or there is a path

${\displaystyle (1)\qquad R=Q_{0}\to Q_{1}\to \cdots \to Q_{m-1}\to Q_{m}=P}$ , with ${\displaystyle m\geq 1}$ ,

of directed edges from ${\displaystyle R}$  to ${\displaystyle P}$ . The vertices forming the path necessarily coincide with the iterated parents ${\displaystyle Q_{j}=\pi ^{j}(R)}$  of ${\displaystyle R}$ , with ${\displaystyle 0\leq j\leq m}$ :

${\displaystyle (2)\qquad R=\pi ^{0}(R)\to \pi ^{1}(R)\to \cdots \to \pi ^{m-1}(R)\to \pi ^{m}(R)=P}$ , with ${\displaystyle m\geq 1}$ ,

In the most important special case (P2) of parents defined as last non-trivial lower central quotients, they can also be viewed as the successive quotients ${\displaystyle R/\gamma _{c+1-j}(R)}$  of class ${\displaystyle c-j}$  of ${\displaystyle R}$  when the nilpotency class of ${\displaystyle R}$  is given by ${\displaystyle c\geq m}$ :

${\displaystyle (3)\qquad R\simeq R/\gamma _{c+1}(R)\to R/\gamma _{c}(R)\to \cdots \to R/\gamma _{c+2-m}(R)\to R/\gamma _{c+1-m}(R)\simeq P}$ , with ${\displaystyle c\geq m\geq 1}$ .

Generally, the descendant tree ${\displaystyle {\mathcal {T}}(G)}$  of a vertex ${\displaystyle G}$  is the subtree of all descendants of ${\displaystyle G}$ , starting at the root ${\displaystyle G}$ . The maximal possible descendant tree ${\displaystyle {\mathcal {T}}(1)}$  of the trivial group ${\displaystyle 1}$  contains all finite p-groups and is somewhat exceptional, since, for any parent definition (P1–P4), the trivial group ${\displaystyle 1}$  has infinitely many abelian p-groups as its immediate descendants. The parent definitions (P2–P3) have the advantage that any non-trivial finite p-group (of order divisible by ${\displaystyle p}$ ) possesses only finitely many immediate descendants.

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 ${\displaystyle j\geq 1}$ , 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 ${\displaystyle \mathrm {cc} (S)=r}$  when the limit ${\displaystyle r=\lim _{j\to \infty }\,\mathrm {cc} (S/\gamma _{j}(S))}$  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 ,^[5] 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

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

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 ^[9] and by C. R. Leedham-Green ,^[10] and conjectured in 1980 already by C. R. Leedham-Green and M. F. Newman.^[2] 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 ${\displaystyle i\geq 1}$  such that the following three conditions are satisfied for any integer ${\displaystyle j\geq i}$ .

S

1. ${\displaystyle \mathrm {cc} (S/\gamma _{j}(S))=r}$ ,
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

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

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

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 ${\displaystyle (R_{n})_{n\geq 0}}$  labelled according to the level ${\displaystyle n}$ , then the finite subtree defined as the difference set

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

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 ${\displaystyle n\geq 0}$ . 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 ${\displaystyle k\geq 0}$  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 ${\displaystyle ({\mathcal {B}}_{k}(n))_{n\geq 0}}$ , resp. branches ${\displaystyle ({\mathcal {B}}(n))_{n\geq 0}}$ , connected by the mainline, whose vertices ${\displaystyle R_{n}}$  are called infinitely capable.

The periodicity of branches of depth-pruned coclass trees has been proved with analytic methods using zeta functions ^[3] of groups by M. du Sautoy ,^[7] and with algebraic techniques using cohomology groups by B. Eick and C. R. Leedham-Green .^[8] 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 ${\displaystyle r\geq 1}$  and dimension ${\displaystyle d}$ , and for any given depth ${\displaystyle k\geq 1}$ , there exists an effective minimal lower bound ${\displaystyle f(k)\geq 1}$ , 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

${\displaystyle (7)\qquad {\mathcal {B}}_{k}(n+d)\simeq {\mathcal {B}}_{k}(n)}$  for all ${\displaystyle n\geq f(k)}$ .

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.

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 ${\displaystyle \mathrm {cc} (G)=r}$ , 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 ^[11] and E. A. O'Brien ^[12] 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 ${\displaystyle \nu (G)=m\geq 2}$ , 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 ${\displaystyle \nu (G)=m\geq 2}$ , 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 ${\displaystyle 1\leq j\leq m-1}$ . Consequently, the descendant tree of ${\displaystyle G}$  is the disjoint union

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

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, ${\displaystyle c=\mathrm {cl} (Q)=\mathrm {cl} (P)+1}$ , from a parent ${\displaystyle P=Q/\gamma _{c}(Q)=\pi (Q)}$  to any immediate descendant ${\displaystyle Q}$ , the coclass remains stable, ${\displaystyle r=\mathrm {cc} (Q)=\mathrm {cc} (P)}$ , if the last non-trivial lower central is cyclic of order ${\displaystyle \vert \gamma _{c}(Q)\vert =p}$ , since then the exponent of the order also increases exactly by a unit, ${\displaystyle \vert Q\vert =p\cdot \vert P\vert }$  . 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 ${\displaystyle \vert \gamma _{c}(Q)\vert =p^{m}}$  with ${\displaystyle m\geq 2}$ . 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

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

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

${\displaystyle (10)\qquad {\mathcal {G}}(p,r)=\left({\dot {\cup }}_{i}\,{\mathcal {T}}(S_{i})\right){\dot {\cup }}{\mathcal {G}}_{0}(p,r)}$ 

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.

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

${\displaystyle \langle \ {\text{order}},\ {\text{counting number}}\ \rangle }$ 

in the following concrete examples of descendant trees, are due to H. U. Besche, B. Eick and E. A. O'Brien .^{[13] [14]} 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

${\displaystyle \langle \ {\text{counting number}}\ \rangle }$ .

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

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

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

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

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 .^[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.

Coclass 0 edit

The coclass graph

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

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 ${\displaystyle \langle 2,1\rangle }$ , for ${\displaystyle p=3}$  it is ${\displaystyle \langle 3,1\rangle }$ .

Coclass 1 edit

The coclass graph

${\displaystyle (12)\qquad {\mathcal {G}}(p,1)={\mathcal {T}}^{1}(R){\dot {\cup }}{\mathcal {G}}_{0}(p,1)}$ 

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 ${\displaystyle \langle 4,2\rangle }$ , resp. ${\displaystyle \langle 9,2\rangle }$ , 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 ${\displaystyle \langle 8,3\rangle }$  and periodicity of length ${\displaystyle 1}$  starting with branch ${\displaystyle {\mathcal {B}}(3)}$ , resp. periodic root ${\displaystyle \langle 81,9\rangle }$  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 ${\displaystyle p\geq 5}$  has unbounded depth and contains non-metabelian groups, and the coclass tree of ${\displaystyle {\mathcal {G}}(p,1)}$  with ${\displaystyle p\geq 7}$  has even unbounded width, that is, the number of descendants of a fixed order increases indefinitely with growing order .^[16]

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 ^[17] for the complete coclass tree ${\displaystyle {\mathcal {T}}^{1}(R)\subset {\mathcal {G}}(p,1)}$ , ${\displaystyle p\in \lbrace 2,3\rbrace }$ , 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 ${\displaystyle s_{j}=\lbrack s_{j-1},x\rbrack }$ , ${\displaystyle 3\leq j\leq n-1=\mathrm {cl} (G)}$ , starting with the main commutator ${\displaystyle s_{2}=\lbrack y,x\rbrack }$ . The nilpotency is formally expressed by the relation ${\displaystyle s_{n}=1}$ , when the group is of order ${\displaystyle \vert G\vert =p^{n}}$ .

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

(13) ${\displaystyle {\begin{aligned}G^{n}(z,w)=&\langle x,y,s_{2},\ldots ,s_{n-1}\mid \\&x^{2}=s_{n-1}^{w},\ y^{2}=s_{2}^{-1}s_{n-1}^{z},\ \lbrack s_{2},y\rbrack =1,\\&s_{2}=\lbrack y,x\rbrack ,\ s_{j}=\lbrack s_{j-1},x\rbrack {\text{ for }}3\leq j\leq n-1\rangle \end{aligned}}}$ 

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

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

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

(14) ${\displaystyle {\begin{aligned}G_{a}^{n}(z,w)=&\langle x,y,s_{2},\ldots ,s_{n-1}\mid \\&x^{3}=s_{n-1}^{w},\ y^{3}=s_{2}^{-3}s_{3}^{-1}s_{n-1}^{z},\ \lbrack y,s_{2}\rbrack =s_{n-1}^{a},\\&s_{2}=\lbrack y,x\rbrack ,\ s_{j}=\lbrack s_{j-1},x\rbrack {\text{ for }}3\leq j\leq n-1\rangle \end{aligned}}}$ 

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 ${\displaystyle r\geq 2}$ , the coclass graph ${\displaystyle {\mathcal {G}}(p,1)}$  exclusively contains p-groups ${\displaystyle G}$  with abelianization ${\displaystyle G/G^{\prime }}$  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 ^[18]).

Coclass 2 edit

The genesis of the coclass graph ${\displaystyle {\mathcal {G}}(p,r)}$  with ${\displaystyle r\geq 2}$  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 ${\displaystyle G/G^{\prime }}$  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 ${\displaystyle p^{3}}$ :

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

Abelianization of type (p,p) edit

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 ${\displaystyle \langle 8,3\rangle }$  is coclass settled, which is the deeper reason for Taussky's Theorem. This remarkable fact has been observed by Giuseppe Bagnera ^[19] in 1898 already.

For odd primes ${\displaystyle p\geq 3}$ , 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 .^[19]

• Firstly, there are two terminal Schur σ-groups ${\displaystyle \langle 243,5\rangle }$  and ${\displaystyle \langle 243,7\rangle }$  in the sporadic part ${\displaystyle {\mathcal {G}}_{0}(3,2)}$  of the coclass graph ${\displaystyle {\mathcal {G}}(3,2)}$ .
• Secondly, the two groups ${\displaystyle G=\langle 243,4\rangle }$  and ${\displaystyle G=\langle 243,9\rangle }$  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 ${\displaystyle \langle 243,3\rangle }$ , ${\displaystyle \langle 243,6\rangle }$  and ${\displaystyle \langle 243,8\rangle }$  give rise to (infinite) coclass trees, e.g., ${\displaystyle {\mathcal {T}}^{2}(\langle 729,40\rangle )}$ , ${\displaystyle {\mathcal {T}}^{2}(\langle 243,6\rangle )}$ , ${\displaystyle {\mathcal {T}}^{2}(\langle 243,8\rangle )}$ , 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 ${\displaystyle d_{2}(G)=\mathrm {dim} _{\mathbb {F} _{p}}(\mathrm {H} ^{2}(G,\mathbb {F} _{p}))}$  coincides with its generator rank ${\displaystyle d_{1}(G)=\mathrm {dim} _{\mathbb {F} _{p}}(\mathrm {H} ^{1}(G,\mathbb {F} _{p}))}$ . A σ-group is a pro-p group ${\displaystyle G}$  which possesses an automorphism ${\displaystyle \sigma \in \mathrm {Aut} (G)}$  inducing the inversion ${\displaystyle x\mapsto x^{-1}}$  on its abelianization ${\displaystyle G/G^{\prime }}$ . A Schur σ-group is a Schur group ${\displaystyle G}$  which is also a σ-group and has a finite abelianization ${\displaystyle G/G^{\prime }}$ .

${\displaystyle \langle 243,3\rangle }$  is not root of a coclass tree,

since its immediate descendant ${\displaystyle \langle 729,40\rangle }$ , which is root of a coclass tree with metabelian mainline vertices, has two siblings ${\displaystyle \langle 729,35\rangle }$ , resp. ${\displaystyle \langle 729,34\rangle }$ , 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 edit

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 ${\displaystyle 2}$  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) ${\displaystyle {\begin{aligned}G(f,g,h)=&\langle a,t,z\mid \\&a^{3}=z^{f},\ \lbrack t,t^{a}\rbrack =z^{g},\ t^{1+a+a^{2}}=z^{h},\\&z^{3}=1,\ \lbrack z,a\rbrack =1,\ \lbrack z,t\rbrack =1\rangle \end{aligned}}}$ 

Abelianization of type (p²,p) edit

For ${\displaystyle p=3}$ , the top levels of the subtree ${\displaystyle {\mathcal {T}}^{2}(\langle 27,2\rangle )}$  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 ${\displaystyle \langle 81,3\rangle }$ , which are of three important kinds.

• Firstly, there are three leaves ${\displaystyle \langle 243,20\rangle }$ , ${\displaystyle \langle 243,19\rangle }$ , ${\displaystyle \langle 243,16\rangle }$  having cyclic centre of order ${\displaystyle 9}$ , and a single leaf ${\displaystyle \langle 243,18\rangle }$  with bicyclic centre of type ${\displaystyle (3,3)}$ .
• Secondly, the group ${\displaystyle G=\langle 243,14\rangle }$  is root of a finite tree ${\displaystyle {\mathcal {T}}(G)={\mathcal {T}}^{2}(G)}$ .
• Finally, the three groups ${\displaystyle \langle 243,13\rangle }$ , ${\displaystyle \langle 243,15\rangle }$  and ${\displaystyle \langle 243,17\rangle }$  give rise to infinite coclass trees, e.g., ${\displaystyle {\mathcal {T}}^{2}(\langle 2187,319\rangle )}$ , ${\displaystyle {\mathcal {T}}^{2}(\langle 243,15\rangle )}$ , ${\displaystyle {\mathcal {T}}^{2}(\langle 243,17\rangle )}$ , 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, ${\displaystyle \langle 243,13\rangle }$  is not root of a coclass tree, since aside from its descendant ${\displaystyle \langle 2187,319\rangle }$ , 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.^[5]

Abelianization of type (p,p,p) edit

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, ${\displaystyle \langle 8,5\rangle }$ , resp. ${\displaystyle \langle 27,5\rangle }$ . This unique tree corresponds to the pro-2 group of the family ${\displaystyle \#59}$  by M. F. Newman and E. A. O'Brien,^[6] 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 edit

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 ${\displaystyle G/G^{\prime }}$  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) edit

Since the elementary abelian p-group ${\displaystyle C_{p}\times C_{p}\times C_{p}}$  of rank ${\displaystyle 3}$ , that is, ${\displaystyle \langle 8,5\rangle }$ , resp. ${\displaystyle \langle 27,5\rangle }$ , 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 ${\displaystyle \langle 32,32\rangle }$  and ${\displaystyle \langle 32,33\rangle }$  are leaves.
• The five groups ${\displaystyle \langle 32,27..31\rangle }$  and the two groups ${\displaystyle \langle 32,34..35\rangle }$  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 ^[6] in the following manner.

${\displaystyle \langle 32,28\rangle }$  gives rise to two trees,

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

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