David J. Green

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Singular

Gap

Cohomology of group number 4 of order 8

About the group

Ring generators

Ring relations

Restriction maps

Back to groups of order 8

General information on the group

• The group is also known as Q8, the Quaternion group of order 8.
• The group has 2 minimal generators and exponent 4.
• It is a (generalized) quaternion group, hence, is of p-Rank 1.
• It 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) · (t2  +  t  +  1)

  (t  −  1) · (t2  +  1)

• The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Benson test.

About the group

Ring generators

Ring relations

Restriction maps

Back to groups of order 8

Ring generators

The cohomology ring has 3 minimal generators of maximal degree 4:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. c_4_0, a Duflot regular element of degree 4

About the group

Ring generators

Ring relations

Restriction maps

Back to groups of order 8

Ring relations

There are 2 minimal relations of maximal degree 3:

1. a_1_1² + a_1_0·a_1_1 + a_1_0²
2. a_1_0³

About the group

Ring generators

Ring relations

Restriction maps

Back to groups of order 8

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. c_4_0 → c_1_0⁴, an element of degree 4

About the group

Ring generators

Ring relations

Restriction maps

Back to groups of order 8