Descendant tree (group theory)

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 ${\displaystyle \langle 3,1\rangle }$.

Figure 2: The coclass graph of finite 2-groups with coclass 1

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 ${\displaystyle \vert G\vert =p^{n}}$.

Figure 3: The coclass graph of finite 3-groups with coclass 1

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]).

Figure 4: The interface between finite 3-groups of coclass 1 and 2 of type (3,3)

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 ${\displaystyle p^{3}}$:

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

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 ${\displaystyle 5}$.

Table 1: Quotients of the groups G=G(f,g,h) [5]

Parameters ${\displaystyle (f,g,h)}$	Abelianization ${\displaystyle G/G^{\prime }}$	Class-2 quotient ${\displaystyle G/\gamma _{3}(G)}$	Class-3 quotient ${\displaystyle G/\gamma _{4}(G)}$	Class-4 quotient ${\displaystyle G/\gamma _{5}(G)}$
${\displaystyle (0,1,0)}$	${\displaystyle (3,3)}$	${\displaystyle \langle 27,3\rangle }$	${\displaystyle \langle 243,3\rangle }$	${\displaystyle \langle 729,40\rangle }$
${\displaystyle (0,1,2)}$	${\displaystyle (3,3)}$	${\displaystyle \langle 27,3\rangle }$	${\displaystyle \langle 243,6\rangle }$	${\displaystyle \langle 729,49\rangle }$
${\displaystyle (1,1,2)}$	${\displaystyle (3,3)}$	${\displaystyle \langle 27,3\rangle }$	${\displaystyle \langle 243,8\rangle }$	${\displaystyle \langle 729,54\rangle }$
${\displaystyle (1,0,0)}$	${\displaystyle (9,3)}$	${\displaystyle \langle 81,3\rangle }$	${\displaystyle \langle 243,15\rangle }$	${\displaystyle \langle 729,79\rangle }$
${\displaystyle (0,0,1)}$	${\displaystyle (9,3)}$	${\displaystyle \langle 81,3\rangle }$	${\displaystyle \langle 243,17\rangle }$	${\displaystyle \langle 729,84\rangle }$
${\displaystyle (0,0,0)}$	${\displaystyle (3,3,3)}$	${\displaystyle \langle 81,12\rangle }$	${\displaystyle \langle 243,53\rangle }$	${\displaystyle \langle 729,395\rangle }$

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

Figure 5: Finite 3-groups of coclass 2 of type (9,3)

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]

Figure 6: Finite 2-groups of coclass 2,3,4 and type (2,2,2)

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 ${\displaystyle (2,2,2)}$.

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 ${\displaystyle p^{4}}$.

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 ${\displaystyle \#74}$.

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

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

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

${\displaystyle \langle 32,34\rangle }$ gives rise to

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