Simon King
David J. Green
Cohomology
→Theory
→Implementation
Jena:
Faculty
External links:
Singular
Gap
|
Cohomology of group number 963 of order 128
General information on the group
- The group has 3 minimal generators and exponent 16.
- It is non-abelian.
- It has p-Rank 2.
- Its center has rank 1.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 2 and depth 2.
- The depth exceeds the Duflot bound, which is 1.
- The Poincaré series is
(t2 + t + 1) · (t6 + t3 + 1) |
| (t − 1)2 · (t2 + 1)2 · (t4 + 1) |
- The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 9 minimal generators of maximal degree 8:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- b_1_2, an element of degree 1
- a_3_3, a nilpotent element of degree 3
- b_4_4, an element of degree 4
- a_5_6, a nilpotent element of degree 5
- b_5_5, an element of degree 5
- a_7_7, a nilpotent element of degree 7
- c_8_9, a Duflot regular element of degree 8
Ring relations
There are 21 minimal relations of maximal degree 14:
- a_1_12 + a_1_0·a_1_1 + a_1_02
- a_1_0·b_1_2
- a_1_03
- b_1_2·a_3_3
- a_1_02·a_3_3
- b_4_4·b_1_2
- b_1_2·a_5_6
- a_3_32 + a_1_0·a_5_6 + b_4_4·a_1_02
- a_1_0·b_5_5
- a_1_02·a_5_6
- a_3_3·b_5_5
- b_1_2·a_7_7
- a_3_3·a_5_6 + a_1_0·a_7_7
- b_4_4·b_5_5 + a_1_02·a_7_7
- a_5_6·b_5_5
- a_5_62 + a_3_3·a_7_7 + b_4_4·a_1_0·a_5_6
- b_5_52 + c_8_9·b_1_22
- a_5_62 + b_4_4·a_1_0·a_5_6 + a_1_02·a_1_1·a_7_7 + c_8_9·a_1_02
- b_5_5·a_7_7
- a_5_6·a_7_7 + b_4_4·a_1_0·a_7_7 + c_8_9·a_1_0·a_3_3
- a_7_72 + c_8_9·a_1_0·a_5_6
Data used for Benson′s test
- Benson′s completion test succeeded in degree 14.
- The completion test was perfect: It applied in the last degree in which a generator or relation was found.
- The following is a filter regular homogeneous system of parameters:
- c_8_9, a Duflot regular element of degree 8
- b_1_24 + b_4_4, an element of degree 4
- The Raw Filter Degree Type of that HSOP is [-1, -1, 10].
- The filter degree type of any filter regular HSOP is [-1, -2, -2].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 1
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_3_3 → 0, an element of degree 3
- b_4_4 → 0, an element of degree 4
- a_5_6 → 0, an element of degree 5
- b_5_5 → 0, an element of degree 5
- a_7_7 → 0, an element of degree 7
- c_8_9 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 2
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- a_3_3 → 0, an element of degree 3
- b_4_4 → 0, an element of degree 4
- a_5_6 → 0, an element of degree 5
- b_5_5 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- a_7_7 → 0, an element of degree 7
- c_8_9 → c_1_04·c_1_14 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 2
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_3_3 → 0, an element of degree 3
- b_4_4 → c_1_14, an element of degree 4
- a_5_6 → 0, an element of degree 5
- b_5_5 → 0, an element of degree 5
- a_7_7 → 0, an element of degree 7
- c_8_9 → c_1_18 + c_1_04·c_1_14 + c_1_08, an element of degree 8
|