David J. Green

Cohomology
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Cohomology of group number 2 of order 81

About the group

Ring generators

Ring relations

Back to groups of order 81

General information on the group

• The group has 2 minimal generators and exponent 9.
• It is abelian, isomorphic to C₉×C₉.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 2.
• The Poincaré series is

  1

  (t  −  1)2

• The a-invariants are -∞,-∞,-2.

About the group

Ring generators

Ring relations

Back to groups of order 81

Ring generators

The cohomology ring has 4 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. c_2_1, an element of degree 2
4. c_2_2, an element of degree 2

About the group

Ring generators

Ring relations

Back to groups of order 81

Ring relations

There are 2 "obvious" relations:
   a_1_0², a_1_1²

Apart from that, there are no relations.

About the group

Ring generators

Ring relations

Back to groups of order 81