Simon King
David J. Green
Cohomology
→Theory
→Implementation
Jena:
Faculty
External links:
Singular
Gap
|
Cohomology of group number 307 of order 128
General information on the group
- The group has 3 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 2.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 3 and depth 3.
- The depth exceeds the Duflot bound, which is 2.
- The Poincaré series is
- The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 5 minimal generators of maximal degree 2:
- a_1_0, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- c_2_4, a Duflot regular element of degree 2
- c_2_5, a Duflot regular element of degree 2
Ring relations
There are 2 minimal relations of maximal degree 2:
- a_1_02
- b_1_1·b_1_2 + a_1_0·b_1_1
Data used for Benson′s test
- Benson′s completion test succeeded in degree 3.
- However, the last relation was already found in degree 2 and the last generator in degree 2.
- The following is a filter regular homogeneous system of parameters:
- c_2_4, a Duflot regular element of degree 2
- c_2_5, a Duflot regular element of degree 2
- b_1_22 + b_1_12, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 3].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 2
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- c_2_4 → c_1_02, an element of degree 2
- c_2_5 → c_1_12, an element of degree 2
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- c_2_4 → c_1_0·c_1_2 + c_1_02, an element of degree 2
- c_2_5 → c_1_12, an element of degree 2
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- c_2_4 → c_1_0·c_1_2 + c_1_02, an element of degree 2
- c_2_5 → c_1_22 + c_1_12, an element of degree 2
|