Mod-5-Cohomology of AlternatingGroup(10), a group of order 1814400

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General information on the group

  • AlternatingGroup(10) is a group of order 1814400.
  • The group order factors as 27 · 34 · 52 · 7.
  • The group is defined by Group([(1,2,3,4,5,6,7,8,9),(8,9,10)]).
  • It is non-abelian.
  • It has 5-Rank 2.
  • The centre of a Sylow 5-subgroup has rank 2.
  • Its Sylow 5-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(400,206); GF(5)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    (1  −  t  +  t2) · (1  −  t  +  t2  −  2·t3  +  2·t4  −  2·t5  +  4·t6  −  2·t7  +  2·t8  −  4·t9  +  2·t10  −  2·t11  +  4·t12  −  2·t13  +  2·t14  −  2·t15  +  t16  −  t17  +  t18)
    ( − 1  +  t)2 · (1  +  t2)2 · (1  +  t4)2 · (1  +  t8)
  • The a-invariants are -∞,-∞,-2. 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, -2].

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 16:

  1. a_6_0, a nilpotent element of degree 6
  2. a_7_0, a nilpotent element of degree 7
  3. a_7_1, a nilpotent element of degree 7
  4. c_8_0, a Duflot element of degree 8
  5. a_15_2, a nilpotent element of degree 15
  6. a_15_3, a nilpotent element of degree 15
  7. c_16_1, a Duflot element of degree 16
  8. c_16_2, a Duflot element of degree 16

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Ring relations

There are 4 "obvious" relations:
   a_7_02, a_7_12, a_15_22, a_15_32

Apart from that, there are 16 minimal relations of maximal degree 32:

  1. a_6_02
  2. a_6_0·a_7_0
  3. a_6_0·a_7_1
  4. a_7_0·a_7_1 + a_6_0·c_8_0
  5. a_6_0·a_15_2
  6. a_6_0·a_15_3
  7. a_7_0·a_15_2 − 2·a_6_0·c_16_2
  8. a_7_0·a_15_3 + a_6_0·c_16_1
  9. a_7_1·a_15_2 − a_6_0·c_16_1
  10. a_7_1·a_15_3 − 2·a_6_0·c_16_2
  11. c_16_2·a_7_0 − 2·c_16_1·a_7_1 + 2·c_8_0·a_15_3
  12. c_16_2·a_7_1 + 2·c_16_1·a_7_0 − 2·c_8_0·a_15_2
  13. a_15_2·a_15_3 + a_6_0·c_8_0·c_16_2
  14. c_16_2·a_15_2 + 2·c_16_1·a_15_3 − c_8_0·c_16_1·a_7_0 + c_8_02·a_15_2
  15. c_16_2·a_15_3 − 2·c_16_1·a_15_2 − c_8_0·c_16_1·a_7_1 + c_8_02·a_15_3
  16. c_16_22 − c_16_12 + c_8_02·c_16_2

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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 32 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_8_0, an element of degree 8
    2. c_16_2, an element of degree 16
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 22].

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Restriction maps

Expressing the generators as elements of H*(SmallGroup(400,206); GF(5))

  1. a_6_0 → a_6_0
  2. a_7_0 → a_7_0
  3. a_7_1 → a_7_1
  4. c_8_0 → c_8_0
  5. a_15_2 → a_15_2
  6. a_15_3 → a_15_3
  7. c_16_1 → c_16_1
  8. c_16_2 → c_16_2

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

  1. a_6_0 → c_2_1·c_2_2·a_1_0·a_1_1, an element of degree 6
  2. a_7_0 → c_2_23·a_1_1 + c_2_13·a_1_0, an element of degree 7
  3. a_7_1 → c_2_1·c_2_22·a_1_0 − c_2_12·c_2_2·a_1_1, an element of degree 7
  4. c_8_0 → c_2_24 + c_2_14, an element of degree 8
  5. a_15_2 → c_2_12·c_2_25·a_1_1 − c_2_15·c_2_22·a_1_0, an element of degree 15
  6. a_15_3 → c_2_13·c_2_24·a_1_0 + c_2_14·c_2_23·a_1_1, an element of degree 15
  7. c_16_1 → c_2_12·c_2_26 − c_2_16·c_2_22, an element of degree 16
  8. c_16_2 → c_2_14·c_2_24, an element of degree 16

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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