David J. Green

Cohomology
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Singular

Gap

Cohomology of group number 1 of order 5

About the group

Ring generators

Ring relations

Back to groups of order 5

General information on the group

• The group has 1 minimal generators and exponent 5.
• It is abelian, isomorphic to C₅.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 1 and depth 1.
• The Poincaré series is

  − 1

  t  −  1

• The a-invariants are -∞,-1.

About the group

Ring generators

Ring relations

Back to groups of order 5

Ring generators

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

1. a_1_0, an element of degree 1
2. c_2_0, an element of degree 2

About the group

Ring generators

Ring relations

Back to groups of order 5

Ring relations

There is one "obvious" relation:
   a_1_0²

Apart from that, there are no relations.

About the group

Ring generators

Ring relations

Back to groups of order 5