David J. Green

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Singular

Gap

Cohomology of group number 9 of order 16

About the group

Ring generators

Ring relations

Restriction maps

Back to groups of order 16

General information on the group

• The group is also known as Q16, the Quaternion group of order 16.
• The group has 2 minimal generators and exponent 8.
• 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 16

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 16

Ring relations

There are 2 minimal relations of maximal degree 3:

1. a_1_0·a_1_1
2. a_1_1³ + a_1_0³

About the group

Ring generators

Ring relations

Restriction maps

Back to groups of order 16

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 16