Mod-2-Cohomology of J1, a group of order 175560

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General information on the group

  • J1 is a group of order 175560.
  • The group order factors as 23 · 3 · 5 · 7 · 11 · 19.
  • The group is defined by Group([(1,262)(2,107)(3,21)(4,213)(5,191)(6,22)(7,133)(8,234)(9,232)(10,151)(11,139)(12,176)(13,202)(14,253)(15,222)(17,195)(18,206)(19,68)(20,55)(23,179)(24,217)(25,216)(26,256)(27,87)(28,70)(29,131)(30,44)(31,105)(32,170)(33,77)(34,104)(35,198)(36,137)(37,243)(38,56)(39,124)(40,223)(41,134)(43,174)(46,51)(47,128)(48,94)(49,250)(50,264)(52,183)(53,231)(54,115)(57,85)(58,233)(59,261)(60,95)(61,235)(62,177)(63,249)(64,91)(65,247)(66,155)(69,219)(71,237)(72,211)(73,84)(74,192)(75,130)(76,251)(79,260)(80,112)(81,193)(82,156)(83,242)(86,238)(88,143)(89,168)(90,148)(92,119)(93,212)(96,150)(97,199)(98,140)(99,189)(100,180)(101,147)(102,111)(103,159)(106,162)(108,194)(109,166)(110,200)(113,120)(114,141)(116,182)(117,181)(118,225)(121,254)(122,125)(123,146)(126,208)(127,221)(129,210)(132,255)(136,175)(138,207)(142,240)(144,172)(145,185)(149,224)(152,169)(153,241)(154,190)(157,214)(158,161)(160,236)(163,239)(164,229)(165,230)(167,188)(171,258)(173,186)(178,245)(184,205)(187,228)(197,203)(201,252)(209,248)(215,259)(218,246)(220,227)(257,263)(265,266),(1,146,21)(2,132,82)(4,156,166)(5,242,253)(6,107,28)(7,125,76)(8,245,130)(9,174,42)(10,241,244)(11,264,63)(12,248,234)(13,36,44)(14,116,128)(15,47,25)(16,178,112)(17,170,110)(18,197,74)(19,233,180)(20,121,96)(22,228,155)(23,48,173)(24,201,187)(26,136,190)(27,212,94)(29,175,52)(30,77,32)(31,237,34)(33,226,90)(35,129,54)(37,161,114)(38,232,87)(39,219,192)(40,78,159)(41,139,71)(43,211,251)(45,222,240)(46,97,135)(49,70,131)(50,153,200)(51,186,209)(53,203,216)(55,169,64)(56,140,230)(57,260,118)(58,91,243)(59,199,227)(60,108,164)(61,208,101)(62,206,106)(65,103,66)(67,95,205)(68,73,225)(69,151,113)(72,221,152)(75,143,202)(79,217,254)(80,93,122)(81,181,252)(83,258,126)(84,163,177)(85,154,213)(86,182,196)(88,133,215)(89,117,247)(92,191,160)(99,229,263)(100,138,188)(102,194,157)(105,149,184)(109,123,193)(111,137,183)(115,238,235)(119,167,147)(120,134,189)(124,185,265)(127,218,261)(141,231,210)(142,239,236)(144,224,249)(145,158,220)(148,214,172)(150,250,259)(162,257,256)(165,179,246)(176,195,266)(198,204,207)(223,262,255)]).
  • It is non-abelian.
  • It has 2-Rank 3.
  • The centre of a Sylow 2-subgroup has rank 3.
  • Its Sylow 2-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(168,43); GF(2)).

General information

  • The cohomology ring is of dimension 3 and depth 3.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1)·((1  −  t  +  t2  −  t3  +  t4) · (1  −  t2  +  t4))
    ( − 1  +  t)3 · (1  +  t  +  t2) · (1  +  t  +  t2  +  t3  +  t4  +  t5  +  t6)
  • The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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Ring generators

The cohomology ring has 5 minimal generators of maximal degree 7:

  1. c_3_0, a Duflot element of degree 3
  2. c_4_0, a Duflot element of degree 4
  3. c_5_0, a Duflot element of degree 5
  4. c_6_1, a Duflot element of degree 6
  5. c_7_1, a Duflot element of degree 7

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Ring relations

There are 2 minimal relations of maximal degree 12:

  1. c_5_02 + c_3_0·c_7_1 + c_4_0·c_6_1 + c_4_0·c_3_02
  2. c_5_0·c_7_1 + c_6_12 + c_6_1·c_3_02 + c_4_03 + c_3_04

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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_3_0, an element of degree 3
    2. c_4_0, an element of degree 4
    3. c_7_1, an element of degree 7
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 11].

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Restriction maps

Expressing the generators as elements of H*(SmallGroup(168,43); GF(2))

  1. c_3_0 → c_3_0
  2. c_4_0 → c_4_0
  3. c_5_0 → c_5_0
  4. c_6_1 → c_6_1
  5. c_7_1 → c_7_1

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 3

  1. c_3_0 → c_1_23 + c_1_12·c_1_2 + c_1_13 + c_1_0·c_1_1·c_1_2 + c_1_0·c_1_12 + c_1_02·c_1_2
       + c_1_03, an element of degree 3
  2. c_4_0 → c_1_24 + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2
       + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2 + c_1_02·c_1_12 + c_1_04, an element of degree 4
  3. c_5_0 → c_1_25 + c_1_14·c_1_2 + c_1_15 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_14
       + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_2 + c_1_05, an element of degree 5
  4. c_6_1 → c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_14·c_1_2
       + c_1_02·c_1_24 + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_14 + c_1_04·c_1_22
       + c_1_04·c_1_1·c_1_2 + c_1_04·c_1_12, an element of degree 6
  5. c_7_1 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7

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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