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Cohomology
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David Green

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Singular

Gap

Mod-3-Cohomology of group number 252 of order 648

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• The group order factors as 2³ · 3⁴.
• It is non-abelian.
• It has 3-Rank 2.
• The centre of a Sylow 3-subgroup has rank 2.
• Its Sylow 3-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

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  −  t3  +  t4  −  t5  +  t6)

  ( − 1  +  t)2 · (1  +  t2)2 · (1  +  t4)

• 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].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 4 minimal generators of maximal degree 8:

1. a_3_0, a nilpotent element of degree 3
2. c_4_0, a Duflot element of degree 4
3. a_7_1, a nilpotent element of degree 7
4. c_8_1, a Duflot element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 2 "obvious" relations:
   a_3_0², a_7_1²

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 10 using the Hilbert-Poincaré criterion.
• However, the last relation was already found in degree 0 and the last generator in degree 8.
• The following is a filter regular homogeneous system of parameters:
  1. c_4_0, an element of degree 4
  2. c_8_1, an element of degree 8
• The above filter regular HSOP forms a Duflot regular sequence.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 10].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(81,2); GF(3))

1. a_3_0 → c_2_2·a_1_1 − c_2_2·a_1_0 − c_2_1·a_1_1 − c_2_1·a_1_0
2. c_4_0 → c_2_2² + c_2_1·c_2_2 − c_2_1²
3. a_7_1 → c_2_1·c_2_2²·a_1_0 + c_2_1²·c_2_2·a_1_1 − c_2_1³·a_1_1 − c_2_1³·a_1_0
4. c_8_1 → c_2_1²·c_2_2² + c_2_1³·c_2_2 + c_2_1⁴

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

1. a_3_0 → 0, an element of degree 3
2. c_4_0 →  − c_2_2² + c_2_1·c_2_2 + c_2_1², an element of degree 4
3. a_7_1 → 0, an element of degree 7
4. c_8_1 → c_2_2⁴ + c_2_1·c_2_2³ + c_2_1²·c_2_2², an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps