Descendant tree (group theory)). He left a 20-volume work. Lin Ding Lin Ding (林鼎) (courtesy name Huanwen (煥文); 891-February 25, 944) was an official of the Chinese Five Dynasties and Ten Kingdoms Period state Wuyue, serving as the chancellor of the state during the reign of its second king Qian Yuanguan (né Qian Chuanguan) and probably during the reign of its third king Qian Hongzuo. Background. Lin Ding was born in 891, during the reign of Emperor Zhaozong of Tang. The family of his father Lin Wuyin (林無隱) was from Houguan (侯官, in modern Fuzhou, Fujian), but Lin Wuyin, who was known 	Lin Ding
27324482	Descendant tree (group theory) Descendant tree (group theory) In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order formula_1, for a fixed prime number formula_2 and varying integer exponents formula_3. Such groups are briefly called finite" "p-groups". The "vertices" of a descendant tree are isomorphism classes of finite "p"-groups. Additionally to their "order" formula_1, finite "p"-groups have two further related invariants, the "nilpotency class" formula_5 and the coclass formula_6. It turned out that descendant trees of a particular kind, the so-called pruned coclass trees whose  infinitely many vertices share a common coclass formula_7, reveal a repeating finite pattern. These two crucial properties of finiteness and periodicity admit a characterization of all members of the tree by finitely many parametrized presentations. 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. An important question is how the descendant tree formula_8 can actually be constructed for an assigned starting group which is taken as the root formula_9 of the tree. The "p"-group generation algorithm is a  recursive process for constructing the descendant tree of a given finite "p"-group playing the role of the tree root. This algorithm is implemented in the computational algebra systems GAP and Magma. Definitions and terminology. According to M. F. Newman, there exist several distinct definitions of the parent formula_10 of a finite "p"-group formula_11. The common principle is to form the quotient formula_12 of formula_11 by a suitable normal subgroup formula_14 which can be either In each case, formula_11 is called an immediate descendant of formula_10 and a "directed edge" of the tree is defined either by formula_33 in the direction  of the canonical projection formula_34 onto the quotient formula_12 or by formula_36 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, by M. du Sautoy and D. Segal, by C. R. Leedham-Green and S. McKay, and by B. Eick, C. R. Leedham-Green, M. F. Newman and E. A. O'Brien. The latter definition is used by M. F. Newman, by M. F. Newman and E. A. O'Brien, by M. du Sautoy, and by B. Eick and C. R. Leedham-Green. In the following , the direction of the canonical projections is selected for all edges. Then, more generally, a vertex formula_9 is a descendant of a vertex formula_38, and formula_38 is an ancestor of formula_9, if either formula_9 is equal to formula_38 or there is a "path" of directed edges from formula_9 to formula_38. The vertices forming the path necessarily coincide with the "iterated parents" formula_47 of formula_9, with formula_49: 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" formula_52 "of class" formula_53 of formula_9 when the  nilpotency class of formula_9 is given by formula_56: Generally, the descendant tree formula_59 of a vertex formula_11 is the subtree of all descendants of formula_11, starting at the root formula_11. The maximal possible descendant tree formula_63 of the trivial group formula_64 contains all finite "p"-groups and is somewhat exceptional, since, for any parent definition (P1–P4), the trivial group formula_64 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 formula_2) possesses only finitely many immediate descendants. 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 formula_67, with formula_68, of the lower central series of a pro-"p" group formula_69 are closed (and open) subgroups of finite index, and therefore the corresponding quotients formula_70 are finite "p"-groups. The pro-"p" group formula_69 is said to be of coclass formula_72 when the limit formula_73 of the coclass of the successive quotients exists and is finite. An infinite pro-"p" group formula_69 of coclass formula_7 is a "p"-adic pre-space group since it has  a normal subgroup formula_76, the "translation group", which is a free module over the ring formula_77 of "p"-adic integers of "uniquely determined" rank formula_29, the "dimension", such that the quotient formula_79 is a finite "p"-group, the "point group", which acts on formula_76 uniserially. The dimension is given by formula_81, with some formula_82. A central finiteness result for infinite pro-"p" groups of coclass formula_7 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 formula_7, for any fixed prime formula_2 and any fixed non-negative integer formula_7. As a consequence, if formula_69 is an infinite pro-"p" group of coclass formula_7, then there exists a minimal integer formula_89 such that the following three conditions are satisfied for any integer formula_90. The descendant tree formula_8, with respect to the parent definition (P2), of the root formula_98 with minimal formula_99 is called the coclass tree formula_100 of formula_69 and its unique maximal infinite (reverse -directed) path formula_102 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 formula_8 with root formula_104 and with mainline vertices formula_105 labelled according to the level formula_106, then the finite subtree defined as the difference set formula_107 is called the "n"th branch (or "twig") of the tree or also the "branch" formula_108 "with root" formula_109, for any formula_3. 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 formula_111 and formula_112 both have  depth formula_113, and the branches formula_114 and formula_115 are pairwise isomorphic as graphs. If all vertices of depth bigger than a given integer formula_116 are removed from the branch formula_117, then we obtain the depth-formula_118 pruned branch formula_119. Correspondingly, the depth-formula_118 pruned coclass tree formula_121, resp. the entire coclass tree formula_8, consists of the infinite sequence of its pruned branches formula_123, resp. branches formula_124, connected by the mainline, whose vertices formula_109 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 formula_69 of coclass formula_127 and dimension formula_29, and for any given depth formula_129, there exists an effective minimal lower bound formula_130, where periodicity of length formula_29 of pruned branches of the coclass tree formula_100 sets in, that is, there exist graph isomorphisms formula_133 for all formula_134. For the proof, click "show" on the right hand side. The graph  isomorphisms of depth-formula_118 pruned branches with roots of sufficiently large order formula_134 are derived with cohomological methods in Theorem 6, p. 277 and Theorem 9, p. 278 by Eick and Leedham-Green and the effective lower bound formula_137 for the branch root orders is established in Theorem 29, p. 287, of this article. 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 formula_138 is called the periodic root of the pruned coclass tree, for a fixed value of the depth formula_118. 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 formula_11 of coclass formula_141, we can distinguish its (entire) descendant tree formula_59 and its coclass-formula_7 descendant tree formula_144, that is the subtree consisting of descendants of coclass formula_7 only. The group formula_11 is called coclass-settled if formula_147, i.e., if there  are no descendants of formula_11 with bigger coclass than formula_7. The nuclear rank formula_150 of formula_11 in the theory of the "p"-group generation algorithm by M. F. Newman and E. A. O'Brien provides the following criteria. In the last case, a more precise assertion is possible: If formula_11 has coclass formula_7 and nuclear rank formula_157, then it gives rise to an "m"-fold multifurcation into a regular coclass-"r" descendant tree formula_144 and formula_163 irregular descendant graphs formula_164 of coclass formula_165, for formula_166. Consequently, the descendant tree of formula_11 is the disjoint union formula_168. 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, formula_169, from a parent formula_170 to any immediate descendant formula_171, the coclass remains stable, formula_172, if the last non-trivial lower central is cyclic of order formula_173, since then the exponent of the order also increases exactly by a unit, formula_174 . In this case, formula_171 is a regular immediate descendant with directed edge formula_176 of step size formula_64, as usual. However, the coclass increases by formula_163, if formula_179 with formula_180. Then formula_171 is called an irregular immediate descendant with directed edge formula_176 of  step size formula_183. If the condition of "step size" formula_64 is imposed on all directed edges, then the maximal descendant tree formula_63 of the trivial group formula_64 splits into a countably infinite disjoint union formula_187 of directed coclass graphs formula_188, which are rather "forests" than trees. More precisely, the above-mentioned Coclass Theorems imply that formula_189 is the disjoint union of "finitely many" coclass trees formula_190 of pairwise non-isomorphic infinite pro-"p" groups formula_191 of coclass formula_7 (Theorem D) and a "finite" subgraph formula_193 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 formula_194 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 formula_195. Depending on the prime formula_2, there is an upper bound on the order of groups for which a SmallGroup identifier exists, e.g. formula_197 for formula_198, and formula_199 for formula_200. 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 formula_64 with its parent formula_38, is denoted by formula_203, and an irregular immediate descendant, connected by an edge of step size formula_204 with its parent formula_38, is denoted by formula_206. 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 formula_207 of finite "p"-groups of coclass formula_113 does not contain any coclass tree and thus exclusively consists of sporadic groups, namely the "trivial group" formula_64 and the "cyclic group" formula_210 of order formula_2, which is a leaf (however, it is capable with respect to the lower exponent-"p" central series). For formula_198 the "SmallGroup identifier" of formula_210 is formula_214, for formula_200 it is formula_216. Coclass 1. The coclass graph formula_217 of finite "p"-groups of coclass formula_64, also called of maximal class, consists of the unique coclass  tree formula_219 with root formula_220, the "elementary abelian" "p"-group "of rank" formula_221, and a single isolated vertex (a terminal orphan without proper parent in the same coclass graph, since the directed edge to the trivial group formula_64 has step size formula_221), the "cyclic group" formula_224 of order formula_225 in the sporadic part formula_226 (however, this group is capable with respect to the lower exponent-"p" central series). The tree formula_227 is the coclass tree of the unique infinite pro-"p" group formula_228 of coclass formula_64. For formula_198, resp. formula_200, the SmallGroup identifier of the root formula_9 is formula_233, resp. formula_234, and a  tree diagram of the coclass graph from branch formula_111 down to branch formula_236 is drawn in Figure 2, resp. Figure 3, where all groups of order at least formula_237 are metabelian, that is non-abelian with derived length formula_221 (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 formula_2 (usually exactly one). The coclass tree  of formula_240, resp. formula_241, has periodic root formula_242 and periodicity of length formula_64 starting with branch formula_244, resp. periodic root formula_245 and periodicity of length formula_221 setting in with branch formula_112. Both trees have branches of bounded depth formula_64, so their virtual periodicity is in fact a strict periodicity. However, the coclass tree of formula_249 with formula_250 has unbounded depth and contains non-metabelian groups, and the coclass tree of formula_249 with formula_252 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 formula_240 and formula_241 of coclass graphs provide an opportunity to give a parametrized polycyclic power-commutator presentation for the complete coclass tree formula_255, formula_256, 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 formula_257 is generated by two elements formula_258 but the presentation contains the series of higher commutators formula_259, formula_260, starting with the  main commutator formula_261. The nilpotency is formally expressed by the relation formula_262, when the group is of order formula_263. For formula_198, there are two parameters formula_265 and the pc-presentation is given by formula_266 The 2-groups of maximal class, that is of coclass formula_64, form three periodic infinite sequences, For formula_200, there are three parameters formula_275 and formula_276 and the pc-presentation is given by formula_277 3-groups with parameter formula_278 possess an abelian maximal subgroup, those with parameter formula_279 do not. More precisely, an existing abelian maximal subgroup is unique, except for the two extra special groups formula_280  and formula_281, where all four maximal subgroups are abelian. In contrast to any bigger coclass formula_282, the coclass graph formula_249 exclusively contains "p"-groups formula_11 with abelianization formula_285 of type formula_286, except for its unique isolated vertex formula_224. The case formula_198 is distinguished by the truth of the reverse statement: Any 2-group with abelianization of type formula_289 is of coclass formula_64 (O. Taussky's Theorem Coclass 2. The genesis of the coclass graph formula_188 with formula_282 is not uniform. "p"-groups with several distinct abelianizations contribute to its constitution. For coclass formula_293, there are essential contributions from groups formula_11 with abelianizations  formula_285 of the types formula_286, formula_297, formula_298, and an isolated contribution by the cyclic group formula_299 of order formula_237: formula_301. Abelianization of type ("p","p"). As opposed to "p"-groups of coclass formula_221 with abelianization of type formula_297 or formula_298, which arise as regular descendants of abelian "p"-groups of the same types, "p"-groups of coclass formula_221 with abelianization of type formula_286 arise from irregular descendants of a non-abelian "p"-group of coclass formula_64 which is not coclass-settled. For the prime formula_198, such groups do not exist at all, since the 2-group formula_242 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 formula_310, the existence of "p"-groups of coclass formula_221 with abelianization of type formula_286 is due to the fact that the group formula_280 is not coclass-settled. Its nuclear rank equals formula_221, which gives rise to a bifurcation of the descendant tree formula_315 into two coclass graphs. The regular component formula_316 is a subtree of the unique tree formula_317 in the coclass graph formula_249. The irregular component formula_319 becomes a subgraph formula_320 of the coclass graph formula_321 when the connecting edges of  step size formula_221 of the irregular immediate descendants of formula_280 are removed. For formula_200, this subgraph formula_325 is drawn in Figure 4, which shows the interface between finite 3-groups with coclass formula_64 and formula_221 of type formula_328. formula_325 has seven top level vertices of three important kinds, all having order formula_330, which have been discovered by G. Bagnera 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 formula_11 whose  relation rank formula_348 coincides with its generator rank formula_349. A σ-group is a pro-"p" group formula_11 which possesses an automorphism formula_351 inducing the inversion formula_352 on its abelianization formula_285. A Schur σ-group is a Schur group formula_11 which is also a σ-group and has a finite abelianization formula_285. formula_340 is not root of a coclass tree, since its immediate descendant formula_357, which is root of a coclass tree with metabelian mainline vertices, has two siblings formula_358, resp. formula_359, which give rise to a single, resp. three, coclass tree(s) with non-metabelian mainline vertices having cyclic centres of  order formula_360 and branches of considerable complexity but nevertheless of bounded depth formula_361. 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 formula_221 having a non-trivial centre of order formula_360. The family members are characterized by three parameters formula_364. Their finite quotients generate all mainline vertices with bicyclic centres of type formula_328 of six coclass trees in the coclass graph formula_334. The association of parameters to the roots of these six trees is given in Table  1, the tree diagrams, except for the abelianization formula_367, are indicated in Figure 4 and Figure 5, and the parametrized pro-3 presentation is given by formula_368 Abelianization of type ("p"²,"p"). For formula_200, the top levels of the subtree formula_370 of the coclass graph formula_334 are drawn in Figure 5. The most important vertices of this tree are the eight siblings sharing the common parent formula_372, which are of three important kinds. Here, formula_381 is not root of a coclass tree, since aside from its descendant formula_390, 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 formula_360 and branches of extreme complexity, here partially even with unbounded depth. Abelianization of type ("p","p","p"). For formula_198, resp. formula_200, there exists a unique coclass tree with "p"-groups of type formula_298 in the coclass graph formula_321. Its root is the elementary abelian "p"-group of type formula_298, that is, formula_397, resp. formula_398. This unique tree corresponds to the pro-2 group of the family formula_399 by M. F. Newman and E. A. O'Brien, resp. to the pro-3 group given by the parameters formula_400 in  Table 1. For formula_198, the tree is indicated in Figure 6, which shows some finite 2-groups with coclass formula_402 of type formula_403. Coclass 3. Here again, "p"-groups with several distinct abelianizations contribute to the constitution of the coclass graph formula_404. There are regular, resp. irregular, essential contributions from groups formula_11 with abelianizations formula_285 of the types formula_407, formula_408, formula_409, formula_410, resp. formula_286, formula_297, formula_298, and an isolated contribution by the cyclic group formula_414 of order formula_415. Abelianization of type ("p","p","p"). Since the elementary abelian "p"-group formula_416 of rank formula_360, that is, formula_397, resp. formula_398, for formula_198, resp. formula_200, is  not coclass-settled, it gives rise to a multifurcation. The regular component formula_422 has been described in the section about coclass formula_221. The irregular component formula_424 becomes a subgraph formula_425 of the coclass graph formula_404 when the connecting edges of step size formula_221 of the irregular immediate descendants of formula_416 are removed. For formula_198, this subgraph formula_325 is contained in Figure 6. It has nine top level vertices of order formula_431 which can be divided into terminal and capable vertices. 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. formula_436 gives rise to two trees, formula_437 associated with family formula_438, and formula_439 associated with family formula_440. formula_441 is associated with family formula_442. formula_443 is associated with family formula_444. formula_445 is associated with family formula_446. formula_447 gives rise to formula_448 associated with family formula_449. Finally, formula_450 is associated with family formula_451. 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 formula_452 of bigger metabelian 2-groups formula_11 of type formula_403 and with  coclass formula_360, 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 formula_221 in formula_11 are denoted by formula_458, for formula_459. 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). Descendant tree (group theory) In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order formula_1, for a fixed prime number formula_2 and varying integer exponents formula_3. Such groups are briefly called "finite" "p-groups". The "vertices" of a descendant tree are isomorphism classes of finite "p"-groups. Additionally to their "order" formula_1, finite "p"-groups have two further related invariants, the "nilpotency class" formula_5 and the coclass formula_6. It turned out that descendant trees of a particular kind, the so