Mod-7-Cohomology of G2(3), a group of order 4245696

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • G2(3) is a group of order 4245696.
  • The group order factors as 26 · 36 · 7 · 13.
  • The group is defined by Group([(2,3)(4,6)(5,7)(8,11)(9,12)(10,14)(13,18)(15,20)(16,21)(17,23)(19,26)(22,30)(24,32)(25,33)(27,36)(28,37)(29,39)(31,42)(34,46)(35,47)(38,51)(40,53)(41,54)(43,57)(44,58)(45,60)(48,64)(49,65)(50,67)(52,70)(55,74)(56,75)(59,79)(61,81)(62,82)(63,84)(66,80)(68,89)(69,90)(71,93)(72,94)(73,96)(76,100)(77,101)(78,103)(83,109)(85,111)(86,112)(87,113)(88,115)(91,108)(92,119)(95,123)(97,125)(98,126)(99,128)(102,124)(104,133)(105,134)(106,135)(107,136)(110,138)(114,143)(117,144)(118,146)(121,148)(122,150)(127,156)(129,158)(130,159)(131,161)(132,163)(137,169)(139,170)(140,171)(141,173)(142,174)(145,178)(147,180)(149,179)(151,182)(152,183)(153,184)(154,185)(155,186)(157,188)(160,191)(162,192)(164,193)(165,194)(166,196)(167,197)(168,199)(172,204)(175,207)(176,208)(177,200)(181,214)(187,220)(189,223)(190,224)(195,229)(198,230)(201,233)(202,234)(203,236)(205,239)(209,242)(211,243)(212,244)(213,245)(215,247)(217,249)(218,231)(219,252)(221,222)(225,256)(226,257)(227,259)(232,264)(235,268)(237,269)(238,270)(241,273)(246,258)(248,280)(250,282)(251,283)(253,285)(254,286)(255,287)(261,279)(262,288)(263,291)(265,292)(266,293)(267,295)(271,299)(272,300)(274,302)(275,303)(276,305)(277,306)(278,307)(281,310)(284,314)(289,318)(290,309)(294,321)(296,308)(297,319)(298,324)(301,328)(304,331)(311,336)(312,337)(313,339)(315,341)(316,342)(317,343)(320,347)(322,349)(323,351)(325,334)(326,353)(327,330)(329,338)(335,359)(340,363)(345,365)(346,352)(348,366)(350,367)(354,370)(355,368)(356,372)(358,369)(360,362)(374,377)(375,376),(1,2,4)(3,5,8)(6,9,13)(7,10,15)(11,16,22)(12,17,24)(14,19,27)(18,25,34)(20,28,38)(21,29,40)(23,31,43)(26,35,48)(30,41,55)(32,44,59)(33,45,61)(36,49,66)(37,50,68)(39,52,71)(42,56,76)(46,62,83)(47,63,85)(51,69,91)(53,72,95)(54,73,97)(57,77,102)(58,78,104)(60,80,106)(64,86,109)(65,87,114)(67,88,116)(70,92,120)(74,98,127)(75,99,129)(79,105,115)(81,107,119)(82,108,137)(84,110,139)(89,117,145)(90,118,147)(93,121,149)(94,122,151)(96,124,153)(100,130,160)(101,131,162)(103,132,164)(111,140,172)(112,141,138)(113,142,175)(123,152,163)(125,154,159)(126,155,161)(128,157,189)(133,165,195)(134,166,148)(135,167,198)(136,168,200)(143,176,209)(144,177,210)(146,179,212)(150,181,215)(156,187,221)(158,190,225)(169,201,207)(170,202,235)(171,203,237)(173,205,236)(174,206,240)(178,211,214)(180,213,246)(182,194,228)(183,216,248)(184,217,250)(185,218,251)(186,219,191)(188,222,254)(192,226,258)(193,227,260)(196,230,262)(197,231,263)(199,232,265)(204,238,271)(208,241,274)(220,253,270)(223,255,288)(224,239,272)(229,261,264)(233,266,294)(234,267,296)(242,275,304)(243,276,256)(244,277,286)(245,278,308)(247,279,309)(249,281,311)(252,284,315)(257,289,302)(259,290,280)(268,297,323)(269,298,325)(273,301,329)(282,312,338)(283,313,340)(285,316,292)(287,317,344)(291,319,346)(293,320,348)(295,322,350)(299,326,354)(300,327,353)(303,330,356)(305,332,314)(306,333,358)(307,334,343)(310,335,360)(318,345,363)(321,339,362)(324,352,369)(328,355,371)(331,357,336)(337,361,347)(341,364,342)(349,351,368)(359,373,367)(365,374,377)(366,375,370)(372,376,378)]).
  • It is non-abelian.
  • It has 7-Rank 1.
  • The centre of a Sylow 7-subgroup has rank 1.
  • Its Sylow 7-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.

Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(42,1); GF(7)).

General information

  • The cohomology ring is of dimension 1 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1)·(1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6  −  t7  +  t8  −  t9  +  t10)
    ( − 1  +  t) · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2) · (1  −  t2  +  t4)
  • The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -1].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 2 minimal generators of maximal degree 12:

  1. a_11_0, a nilpotent element of degree 11
  2. c_12_0, a Duflot element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There is one "obvious" relation:
   a_11_02

Apart from that, there are no relations.

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 12 using the Hilbert-Poincaré criterion.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_12_0, an element of degree 12
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, 11].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(42,1); GF(7))

  1. a_11_0 → a_11_0
  2. c_12_0 → c_12_0

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

  1. a_11_0 → c_2_05·a_1_0, an element of degree 11
  2. c_12_0 → c_2_06, an element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps


Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2

Last change: 14.12.2010