Periodic structures in the graph associated with p-groups of maximal class [Elektronische Ressource] / von Heiko Dietrich
125 Pages
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Periodic structures in the graph associated with p-groups of maximal class [Elektronische Ressource] / von Heiko Dietrich

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125 Pages
English

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Periodic structures in the graph associated with p-groups of maximalclassVon der Carl-Friedrich-Gauß-Fakult¨at derTechnischen Universit¨at Carolo-Wilhelmina zu Braunschweigzur Erlangung des GradesDoktor der Naturwissenschaften (Dr. rer. nat.)genehmigte Dissertation vonHeiko Dietrichgeboren am 15. April 1980in Langen (Hessen)Eingereicht am: 27.04.2009Mu¨ndliche Pru¨fung am: 26.06.2009Referentin: Prof. Dr. Bettina EickKorreferent: Prof. Dr. Gerhard Hiß2009iiDedicated to my grandfatherHans Viktor KleinivSummaryA finite group whose order is a power of a prime p is called a finite p-group. Among finitegroups, p-groups take a special position: For example, every finite group contains large p-groups as subgroups by the Sylow Theorems. The classification of p-groups is a difficultproblem and, in general, not even the exact number of isomorphism types of groups of ordernp is known. The asymptotic estimates of Higman (1960) and Sims (1965) show that there3 8/32n /27+O(n ) nare p isomorphism types of groups of order p . Higman’s PORC Conjecturen(1960) claims that for fixed n the number of isomorphism types of groups of order p is apolynomial on residue classes.A special type ofp-groups arep-groups of maximal class: These are thep-groups of ordernp with nilpotency class n−1. A first major study of maximal class groups was carriedout by Blackburn in 1958.

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Periodic structures in the graph associated with p-groups of maximal
class
Von der Carl-Friedrich-Gauß-Fakult¨at der
Technischen Universit¨at Carolo-Wilhelmina zu Braunschweig
zur Erlangung des Grades
Doktor der Naturwissenschaften (Dr. rer. nat.)
genehmigte Dissertation von
Heiko Dietrich
geboren am 15. April 1980
in Langen (Hessen)
Eingereicht am: 27.04.2009
Mu¨ndliche Pru¨fung am: 26.06.2009
Referentin: Prof. Dr. Bettina Eick
Korreferent: Prof. Dr. Gerhard Hiß
2009iiDedicated to my grandfather
Hans Viktor KleinivSummary
A finite group whose order is a power of a prime p is called a finite p-group. Among finite
groups, p-groups take a special position: For example, every finite group contains large
pgroups as subgroups by the Sylow Theorems. The classification of p-groups is a difficult
problem and, in general, not even the exact number of isomorphism types of groups of order
np is known. The asymptotic estimates of Higman (1960) and Sims (1965) show that there
3 8/32n /27+O(n ) nare p isomorphism types of groups of order p . Higman’s PORC Conjecture
n(1960) claims that for fixed n the number of isomorphism types of groups of order p is a
polynomial on residue classes.
A special type ofp-groups arep-groups of maximal class: These are thep-groups of order
np with nilpotency class n−1. A first major study of maximal class groups was carried
out by Blackburn in 1958. He obtained a classification of the 2- and 3-groups of maximal
class, and he observed that a classification for primes greater than 3 is significantly more
difficult. Following Blackburn, maximal class groups were discussed in detail by Shepherd
(1971), Miech (1970 – 1982), Leedham-Green & McKay (1976 – 1984), Ferna´ndez-Alcober
(1995), and Vera-L´opez et al. (1995 – 2008). Despite substantial progress made in the last
six decades, the classification of maximal class groups is still an open problem in p-group
theory. For example, Problem 3 of Shalev’s survey paper (1994) on finite p-groups asks to
classify the 5-groups of maximal class.
nThe coclass of ap-group of orderp and nilpotency classc is defined asn−c. Hence, the
p-groups of maximal class are thep-groups of coclass 1. Leedham-Green & Newman (1980)
suggested to classify p-groups by coclass, and their suggestion has led to a major research
project in p-group theory. In this thesis, we follow the philosophy of coclass theory and
investigate the p-groups of maximal class (or coclass 1).
The graphG(p). The coclass graphG(p) associated with p-groups of maximal class is
defined as follows: Its vertices are the isomorphism types of finitep-groups of maximal class
where a vertex is identified with a group representing its isomorphism class. Two vertices
G and H are connected by a directed edge G→ H if and only if G is isomorphic to the
central quotient H/ζ(H). A group H is called a descendant of a group G inG(p) if G =H
or if there is a path fromG toH. We visualizeG(p) in the Euclidean plane as an undirected
graph by drawing the proper descendants of a group inG(p) below that group.
2It is well-known thatG(p) consists of the cyclic group of order p and an infinite tree
2T(p), whose root is elementary abelian of orderp . The treeT(p) contains a unique infinite
path starting at its root. This path is called the mainline ofT(p) and we denote it by
nS → S → ... where S has order p . The n-th branchB of the treeT(p) is the finite2 3 n n
subtree ofT(p) induced by the descendants of S which are not descendants of S . Asn n+1
usual, the depth of a subtreeB ofT(p) is the maximal length of a path withinB, and its
width is the maximal number of vertices at the same depth inB. A sketch ofG(p) is given
in Figure 1.
vvi
S2 C 2pS3
S4 B2
B3
B4
Figure 1: The graphG(p).
By the results of Blackburn, the graphG(p) for p = 2,3 is completely understood. For
example, all branches have depth 1 and a recurrent structure. Leedham-Green & McKay
(1984) showed that the set of depths of the branches ofT(p) is unboundedforp≥5 and the
set of widths of the branches ofT(5) is bounded. Here we prove that the set of widths of
the branches ofT(p) is unboundedforp≥7. Forp=5 an investigation with computational
methods is still possible, see Newman (1990) and Dietrich, Eick & Feichtenschlager (2008),
whereas for p≥ 7 the size of the branches increases too fast for a complete examination.
The detailed structure ofG(p) for p≥7 is not known.
Theperiodicity of type1. Wenowdescribeourfirstmainresultsand,forthispurpose,
we introduce some more notation. For k≥ 0 the shaved branchB [k] is the subtree ofBn n
induced by the groups of distance at most k from its root S . We define functions c = c(p)n
and e = e (p), both essentially given by linear polynomials, which for given p satisfyn n
0≤ e ≤ e ≤... and e = e +d whered =p−1. We call the shaved branchT =B [e ]2 3 n+d n n n n
the n-th body ofT(p) and prove the following for n≥p+1.
• The depths ofB andT differ by at most c.n n
• There is an embeddingι=ι :T ֒→B of rooted trees such that ι(T ) =B [e ].n n n+d n n+d n
A summary of these results is visualized in Figure 2.
Sn
Sn+d
Tι nSn+2d
enTι n+d ≤c
en BndeT n+dn+2d ≤c
en+d Bn+dden+2d
≤c
Bn+2d
Figure 2: The periodicity of type 1.
This recurrent pattern inT(p) is referred to as the periodicity of type 1. It shows that a
major part of the tree carries a periodic structure. Weaker versions of this periodic pattern
inT(p) have been proved by Eick & Leedham-Green (2008) and du Sautoy (2001).
Experiments by computer suggest that the whole branchB cannot be embedded inton
B . The periodicity of type 1 as proved in this thesis embedsB with the exclusion of atn+d n
most c levels of groups. This shows that the periodicity of type 1 is close to best possible.vii
The periodicity of type 2. For a complete description of the treeT(p) it is necessary
to describe the difference graph ofB and ι(T ) =B [e ]. Since in general the set ofn nn+d n+d
widths of the branches is unbounded, this graph cannot be isomorphic to a subgraph ofB .n
However, a conjecture of Eick, Leedham-Green, Newman & O’Brien (2009) claims that it
can be described by another periodic pattern, that is, a periodicity of type 2. According
to this conjecture, for large enough n, the subtree ofB induced by the descendants ofn+d
a group at depth e inB is isomorphic to a corresponding subtree inB . Thus, then n+d n
periodicities of type 1 and 2 would in principle suffice to describe the treeT(p) completely.
As a first approximation of this conjecture, we consider the difference graph ofT andn+d
ι(T ) and, thus, omit at most c levels of groups. We define thed-step descendant treeD (G)n d
of a groupG inT(p) as the subtree ofT(p) induced by the descendants of distance at most
d from G. Then the periodicity of type 2 asserts that for large enough n every group G
at depth e inT has a periodic parent H at depth e −d inT such thatD (H) andn n+d n n+d d
D (G) are isomorphic as rooted trees, see Figure 3.d
Sn+d
Hen−d
Gen
∼=en+d
Tn+d
Figure 3: A periodic parent of G.
This periodic pattern and the periodicity of type 1 would suffice to describe the bodies
ofT(p) completely. The main problem is to specify a periodic parent of a group. Based
on ideas of Leedham-Green & McKay (1984), we use p-adic number theory and prove the
periodicity of type 2 in certain special cases. There are significant differences depending on
the residue p modulo 6, and we consider the easier case p≡ 5 mod 6 here. As a corollary,
we show that for large enough n the d-step parent of a group at depth e inT whichn n+d
has proper descendants is a periodic parent if its automorphism group is a p-group. The
computation of explicit examples indicate that the d-step parent is not always a periodic
parent, and we propose an alternative construction of periodic parents in a special case.
Classification of groups. The periodicities of type 1 and 2 describe graph theoretic
patterns withinT(p). For our proof of these periodicities we use a cohomological approach
and describe the groups in the bodies ofT(p) as certain group extensions. This allows us to
construct all graph isomorphisms on a group theoretic level such that the periodic patterns
inT(p) are reflected in the structure of the groups involved. For example, if G is a group
inT with n≥ p+1, then a suitable choice of the embeddings ι allows us to describe then
2infinitely many groups in{G,ι(G),ι (G),...} by a single group presentation whose defining
relations contain one indeterminate integer as parameter.
The 5-groups of maximal class. As an application, we prove that the bodiesT =n
B [n−4] of the treeT(5) can be described by a finite subgraph and the periodicities of typen
1 and 2. We deduce that the infinitely many groups in these bodies can be described by
finitely many group presentations with at most two indeterminate integers as parameters.
This is close to a positive answer of Problem 3 in Shalev’s survey paper (1994).viiiZusammenfassung
nEine endliche p-Gruppe ist eine Gruppe mit Primzahlpotenzordnung p . In der Klasse
der endlichen Gruppen nehmen p-Gruppen eine besondere Stellung ein: Beispielsweise
folgt aus den Sylow-Sa¨tzen, dass jede endliche Gruppe große p-Gruppen als Untergruppen
entha¨lt. Die Klassifikation von p-Gruppen ist ein schwieriges Unterfangen und im
Allgemeinen ist nicht einmal die exakte Anzahl der Isomorphietypen von Gruppen der Ordnung
np bekannt. AsymptotischeAbscha¨tzungen von Higman(1960) undSims(1965) zeigen, dass
3 8/32n /27+O(n ) nesp Isomorphietypen von Gruppender Ordnungp gibt. Higmans PORC
Vermutung(1960) sagt voraus, dassdie Anzahlder Isomorphietypen von GruppenderOrdnung
np fu¨r festes n ein Polynom auf Restklassen ist.
Ein besonderer Typ von p-Gruppen sind die p-Gruppen mit maximaler Klasse. Dies
nsind die p-Gruppen der Ordnung p mit Nilpotenzklasse n− 1. Eine erste grundlegende
Untersuchung von Gruppen mit maximaler Klasse wurde 1958 von Blackburn
vorgenommen. Blackburn erzielte die Klassifikation der 2- und 3-Gruppen mit maximaler Klasse
und er beobachtete, dass eine Klassifikation fu¨r gro¨ßere Primzahlen weitaus schwieriger ist.
Weitere detaillierte Untersuchungen der Gruppen mit maximaler Klasse wurden von
Shepherd (1971), Miech (1970 – 1982), Leedham-Green & McKay (1976 – 1984),
Ferna´ndezAlcober (1995) und Vera-L´opez et al. (1995 – 2008) durchgefu¨hrt. Trotz erheblichen
Fortschritts in den letzten sechs Jahrzehnten ist die Klassifikation der Gruppen mit
maximaler Klasse noch immer ein offenes Problem in der Theorie derp-Gruppen. Beispielsweise
¨fragt Shalev (1994) in Problem 3 seines Ubersichtsartikels u¨berp-Gruppen nach einer
Klassifikation der 5-Gruppen mit maximaler Klasse.
nDie Koklasse einerp-Gruppe mit Ordnungp und Nilpotenzklassec ist definiert alsn−c,
das heißt, p-Gruppen mit maximaler Klasse entsprechen den p-Gruppen mit Koklasse 1.
Leedham-Green & Newman (1980) machten den Vorschlag, p-Gruppen nach ihrer Koklasse
zu klassifizieren, und legten damit den Grundstein fu¨r ein umfangreiches Forschungsprojekt
in der Theorie der p-Gruppen. In der vorliegenden Arbeit wird diese Theorie benutzt, um
diep-Gruppen mit maximaler Klasse (oder Koklasse 1) zu untersuchen.
Der Graph G(p). Der den p-Gruppen mit maximaler Klasse zugeordnete
KoklassengraphG(p) ist wie folgt definiert: Die Knoten sind die Isomorphietypen von endlichen
pGruppenmitmaximaler Klasse, wobeiein Knoten miteinemIsomorphietyp-Repr¨asentanten
identifiziertwird. ZweiKnotenGundH sindgenaudannmiteinergerichtetenKanteG→H
verbunden,wennGisomorphzudemzentralenQuotientenH/ζ(H)ist. EineGruppeH heißt
Nachfolger einer GruppeG inG(p), fallsG =H oder falls es einen Pfad vonG nachH gibt.
Der gerichtete GraphG(p) wirdin der EuklidischenEbeneungerichtet dargestellt, indemdie
echten Nachfolger einer Gruppe inG(p) unterhalb dieser Gruppe gezeichnet werden. Es ist
2bekannt, dass sichG(p) aus der zyklischen Gruppe der Ordnungp und einem unendlichen
2BaumT(p)mitelementar-abelscher WurzelderOrdnungp zusammensetzt. DieWurzelvon
T(p)istderStartknoteneineseindeutigenunendlichenPfades. DieserPfadistdieHauptlinie
nvonT(p) und wird als S →S → ... bezeichnet, wobei S die Ordnung p hat. Der n-te2 3 n
ixx
AstB des BaumesT(p) ist der endliche Teilbaum vonT(p), der von den Nachfolgern vonn
S induziert wird, welche nicht auch Nachfolger von S sind. Wie u¨blich sind Tiefe undn n+1
Weite eines Wurzelbaumes definiert als die maximale La¨nge eines Pfades, beziehungsweise
die maximale Anzahl von Knoten der gleichen Tiefe. In Abbildung 1 ist die Struktur von
G(p) skizziert.
S2 C 2pS3
S4 B2
B3
B4
Abbildung 1: Der GraphG(p).
Die Resultate von Blackburn beschreiben die GraphenG(p) fu¨rp=2,3 vollsta¨ndig und es
¨ist bekannt, dassalle Aste die Tiefe 1 undperiodisch auftretende Strukturhaben.
Leedham¨Green&McKay(1984)habengezeigt, dassfu¨rp≥5dieMengederTiefenderAstevonT(p)
¨unbeschra¨nkt und die Menge der Weiten der Aste vonT(5) beschra¨nkt ist. In der
vorliegen¨den Arbeit wird bewiesen, dass die Menge der Weiten der Aste vonT(p) unbeschra¨nkt ist
fu¨rp≥7. W¨ahrendT(5)daher detailliert mitdemComputeruntersuchtwerdenkann, siehe
zumBeispiel Newman (1990) undDietrich, Eick &Feichtenschlager (2008), soisteine solche
¨ausfu¨hrliche Untersuchung fu¨rp≥7 auf Grund des Wachstums der Aste nicht mo¨glich. Die
detaillierte Struktur vonG(p) fu¨rp≥7 ist daher unbekannt.
DiePeriodizit¨at vomTyp1. UmdasersteHauptresultatdieserArbeitzubeschreiben,
ist weitere Notation no¨tig. Fu¨r k≥ 0 ist der gestutzte AstB [k] der Teilbaum vonB ,n n
welcher von den Gruppen der Tiefe ho¨chstens k inB induziert wird. Weiterhin werdenn
Abbildungen, im Wesentlichen lineare Polynome, c = c(p) und e = e (p) definiert mitn n
0≤ e ≤ e ≤... und e = e +d fu¨r festes p und d =p−1. Der n-te Rumpf vonT(p)2 3 n+d n
ist der gestutzte AstT =B [e ]. Folgende Aussagen werden fu¨rn≥p+1 bewiesen:n n n
• Die Tiefen vonB undT unterscheiden sich ho¨chstens um c.n n
• Es gibt eine Einbettungι =ι :T ֒→B von Wurzelb¨aumen mit ι(T )=B [e ].n n n+d n n+d n
Eine Zusammenfassung dieser Ergebnisse ist in Abbildung 2 skizziert.
Sn
Sn+d
ι TnSn+2d
enι Tn+d ≤c
en BndeT n+dn+2d ≤c
en+d Bn+dden+2d
≤c
Bn+2d
Abbildung 2: Die Periodizita¨t vom Typ 1.