Rossmann_Periodicities

8/14/2019 Rossmann_Periodicities

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Periodicities in trees associatedwith the classication of

p-groups by width, rank andobliquity

Tobias Romann

Report on Diploma Thesis

p-Groups: Rank, Width, and Obliquity

supervised by Professor Dr. Bettina Eick

TU Braunschweig

11 July 2008


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Introduction

Denition. Let G be a pro- p group.

The rank of G issup{d(U ) | U c G}.

The width of G issupi 1

log p (

| i (G) : i +1 (G)

|).

The obliquity of G issupi 1

log p (| i (G) : i (G)|)

where

i (G) = {N c G | N < i (G)}.


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The graph G( p,r,w,o ) Vertices correspond to nite p-groups of

rank r , width w, and obliquity o (up to

isomorphism).

An edge joins G/ c (G) to G where c is theclass of G.

Project: Analyse

G( p,r,w,o ).

(1) Determine the innite paths. Classify innite pro- p groups of rank r ,width w, and obliquity o up toisomorphism.

(2) Investigate the branches attached to theinnite paths. Is there a periodicity?

(3) Consider the remaining sporadic groups.


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Innite pro- p groups

Klaas et al.:

Classication of innite pro- p groups of nite rank, width, and obliquity up toisomorphism of their Lie algebras.

The possible Lie algebras correspond tomaximal groups.

Each innite pro- p group of nite rank,width, and obliquity can be obtained as asubgroup of one of these maximal groups.

Example:

There are two simple three-dimensional Liealgebras over Q p : sl 2 (Q p ) and sl 1 (D ).

Let G and H be the correspondingmaximal groups.

Leedham-Green & McKay: If p 3, thenG and H represent the two isomorphismclasses of innite pro- p groups of rank 3,width 2, and obliquity 0.


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G( p, 3, 2, 0) for p 3 Sanderson, Leedham-Green & McKay:

First investigation of the graph. Inparticular, they proved that branches havedepth at most 2 if p 5.

Detailed analysis using GAP , in particularANUPQ . Result: Branches seem to be periodic with

period 2.

Precise shapes . . .


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B 2 j (G)

G2 j

G2 j +1

G2 j +2

. . .

. . .

p times

G2 j +1

G2 j +2

( j 2, p 5)


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B 2 j (H )

H 2 j

H 2 j +1

H 2 j +2

. . .

. . .

p 1 times

H 2 j +1

H 2 j +2

( j 2, p 5)


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B 2 j (G)G2 j

G2 j +1

G2 j +2

G2 j +3

G2 j +4

4

3

2 2 2

( j 2, p = 3)

G2

G2


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B 2 j (H ) ( j 2) B 2 j 1 (H )

H 2 j

H 2 j +1

H 2 j +2

5

H 2 j 1

H 2

j

H 2 j +1

H 2 j +2

5 5

( p = 3)


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sl n (K )

Another interesting class of pro- p groups:The maximal groups with Lie algebra of the form sl n (K ).

Investigate the associated branches;ignore the rank, width, and obliquity andconsider only those groups with the rightlower central pattern.

An example: sl 2 (K ) for p = 3 andK = Q 3 (3).

Result: Graph seems to be periodic withperiod 4.

Explicit branches . . .


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Notation

n k n n . . .

. . .

k times


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sl 2(K ), p = 3 , K = Q 3(

3)

15 8 27 6

105

Class

6

7

8

4

27 20

165

186 369

6

Class

7

8

9

10

11


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sl 2(K ), p = 3 , K = Q 3(

3)

15 4 18 4 27 6

87

Class

8

9

10

5

15 8 27 16

200

189 369

7

Class

9

10

11

12

13


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sl n (K )

Klaas et al.: classication of maximalgroups of dimension 14.

Similar results occur for various choices of p, elds K with

|K : Q p

| 4, and n

3.

As these numbers increase, the groupsbecome less accessible.

(partial) data support the conjecture thatthese branches always repeat periodically


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Cohomology and

G( p, 3, 2, 0)

Let p 5 and Q be either G or H . A group in Bi (Q) is an extension of a

Q2 -module A by Q i .

Using the results of Sanderson, A is foundto be a Q2 -module quotient of 2 (Q)/ 4 (Q) or of 3 (Q)/ 5 (Q).

Theorem. H 2 (Q i , A)= H 2 (Q i +2 , A) for all

sufficiently large i.

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