David J. Green

Cohomology
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Cohomology of group number 5 of order 27

About the group

Ring generators

Ring relations

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General information on the group

• The group has 3 minimal generators and exponent 3.
• It is abelian, isomorphic to C₃×C₃×C₃.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 3.
• The Poincaré series is

  − 1

  (t  −  1)3

• The a-invariants are -∞,-∞,-∞,-3.

About the group

Ring generators

Ring relations

Back to groups of order 27

Ring generators

The cohomology ring has 6 minimal generators of maximal degree 2:

1. a_1_0, an element of degree 1
2. a_1_1, an element of degree 1
3. a_1_2, an element of degree 1
4. c_2_3, an element of degree 2
5. c_2_4, an element of degree 2
6. c_2_5, an element of degree 2

About the group

Ring generators

Ring relations

Back to groups of order 27

Ring relations

There are 3 "obvious" relations:
   a_1_0², a_1_1², a_1_2²

Apart from that, there are no relations.

About the group

Ring generators

Ring relations

Back to groups of order 27