Mathematics:

Cohomology
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Jena:

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David Green

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Singular

Gap

Mod-19-Cohomology of group number 7 of order 342

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• The group order factors as 2 · 3² · 19.
• It is non-abelian.
• It has 19-Rank 1.
• The centre of a Sylow 19-subgroup has rank 1.
• Its Sylow 19-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.

Structure of the cohomology ring

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) · (1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6) · (1  +  t  −  t5  −  t6  −  t7  −  t8  +  t10  +  t11  +  t12  +  t13  +  t14  −  t16  −  t17  −  t18  −  t19  +  t23  +  t24))

  ( − 1  +  t) · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2) · (1  −  t2  +  t4) · (1  −  t3  +  t6) · (1  +  t3  +  t6) · (1  −  t6  +  t12)

• 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 36:

1. a_35_0, a nilpotent element of degree 35
2. c_36_0, a Duflot element of degree 36

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There is one "obvious" relation:
   a_35_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 36 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_36_0, an element of degree 36
• The above filter regular HSOP forms a Duflot regular sequence.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, 35].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(19,1); GF(19))

1. a_35_0 → c_2_0¹⁷·a_1_0
2. c_36_0 → c_2_0¹⁸

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

1. a_35_0 → c_2_0¹⁷·a_1_0, an element of degree 35
2. c_36_0 → c_2_0¹⁸, an element of degree 36

About the group

Ring generators

Ring relations

Completion information

Restriction maps