Simon King
David J. Green
Cohomology
→Theory
→Implementation
Jena:
Faculty
External links:
Singular
Gap
|
Cohomology of group number 2001 of order 128
General information on the group
- The group has 4 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 2.
- It has 4 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 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t7 + 2·t6 + 2·t5 + t3 + 2·t2 + 2·t + 1) |
| (t + 1) · (t − 1)3 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-6,-3. 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:
- b_1_0, an element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- b_1_3, an element of degree 1
- c_2_8, a Duflot regular element of degree 2
- b_5_26, an element of degree 5
- b_5_27, an element of degree 5
- b_5_28, an element of degree 5
- c_8_61, a Duflot regular element of degree 8
Ring relations
There are 16 minimal relations of maximal degree 10:
- b_1_0·b_1_2
- b_1_1·b_1_3 + b_1_0·b_1_1
- b_1_22·b_1_3 + b_1_1·b_1_22 + b_1_13 + b_1_02·b_1_3 + b_1_03
- b_1_04·b_1_3 + b_1_05
- b_1_3·b_5_26 + b_1_0·b_5_26
- b_1_0·b_5_27
- b_1_2·b_5_26 + b_1_1·b_5_27 + b_1_1·b_1_25 + b_1_15·b_1_2 + c_2_8·b_1_1·b_1_23
+ c_2_82·b_1_1·b_1_2
- b_1_3·b_5_27 + b_1_2·b_5_28 + c_2_8·b_1_1·b_1_23
- b_1_1·b_5_28 + b_1_04·b_1_12 + c_2_8·b_1_14 + c_2_8·b_1_03·b_1_1
+ c_2_82·b_1_0·b_1_1
- b_1_22·b_5_28 + b_1_1·b_1_2·b_5_27 + b_1_12·b_5_26 + b_1_02·b_5_28 + b_1_05·b_1_12
+ b_1_06·b_1_1 + c_2_8·b_1_1·b_1_24 + c_2_8·b_1_15 + c_2_8·b_1_05 + c_2_82·b_1_13 + c_2_82·b_1_03
- b_5_26·b_5_28 + b_1_04·b_1_1·b_5_26 + b_1_05·b_5_28 + b_1_09·b_1_1
+ c_2_8·b_1_13·b_5_26 + c_2_8·b_1_03·b_5_26 + c_2_8·b_1_08 + c_2_82·b_1_0·b_5_26 + c_2_82·b_1_06
- b_5_272 + b_1_25·b_5_27 + b_1_02·b_1_33·b_5_28 + b_1_05·b_5_28 + c_8_61·b_1_22
+ c_2_8·b_1_23·b_5_27 + c_2_82·b_1_26 + c_2_83·b_1_14
- b_5_26·b_5_27 + b_1_15·b_5_27 + c_8_61·b_1_1·b_1_2 + c_2_82·b_1_1·b_5_27
+ c_2_82·b_1_1·b_1_25 + c_2_83·b_1_13·b_1_2
- b_5_27·b_5_28 + b_1_2·b_1_34·b_5_28 + b_1_1·b_1_24·b_5_27 + b_1_15·b_5_27
+ c_8_61·b_1_2·b_1_3 + c_2_8·b_1_13·b_5_27 + c_2_82·b_1_1·b_1_25 + c_2_82·b_1_15·b_1_2
- b_5_282 + b_1_35·b_5_28 + b_1_05·b_5_28 + b_1_08·b_1_12 + c_8_61·b_1_32
+ c_8_61·b_1_02 + c_2_8·b_1_02·b_1_3·b_5_28 + c_2_8·b_1_03·b_5_28 + c_2_82·b_1_16 + c_2_82·b_1_0·b_1_35 + c_2_82·b_1_02·b_1_34 + c_2_82·b_1_06 + c_2_83·b_1_03·b_1_3 + c_2_83·b_1_04 + c_2_84·b_1_02
- b_5_262 + b_1_15·b_5_26 + b_1_04·b_1_1·b_5_26 + b_1_08·b_1_12 + b_1_09·b_1_1
+ b_1_010 + c_8_61·b_1_12 + c_2_8·b_1_13·b_5_26 + c_2_8·b_1_18 + c_2_8·b_1_03·b_5_28 + c_2_8·b_1_07·b_1_1 + c_2_82·b_1_06 + c_2_83·b_1_04 + c_2_84·b_1_12
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- 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_2_8, a Duflot regular element of degree 2
- c_8_61, a Duflot regular element of degree 8
- b_1_32 + b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 9].
- 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
- b_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
- b_1_3 → 0, an element of degree 1
- c_2_8 → c_1_02, an element of degree 2
- b_5_26 → 0, an element of degree 5
- b_5_27 → 0, an element of degree 5
- b_5_28 → 0, an element of degree 5
- c_8_61 → c_1_18 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_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
- b_1_3 → c_1_2, an element of degree 1
- c_2_8 → c_1_02, an element of degree 2
- b_5_26 → 0, an element of degree 5
- b_5_27 → 0, an element of degree 5
- b_5_28 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_04·c_1_2, an element of degree 5
- c_8_61 → c_1_12·c_1_26 + c_1_18 + c_1_04·c_1_24 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_1_3 → 0, an element of degree 1
- c_2_8 → c_1_0·c_1_2 + c_1_02, an element of degree 2
- b_5_26 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- b_5_27 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- b_5_28 → c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
- c_8_61 → c_1_12·c_1_26 + c_1_18 + c_1_0·c_1_27 + c_1_0·c_1_12·c_1_25
+ c_1_0·c_1_14·c_1_23 + c_1_02·c_1_26 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_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
- b_1_3 → 0, an element of degree 1
- c_2_8 → c_1_0·c_1_2 + c_1_02, an element of degree 2
- b_5_26 → 0, an element of degree 5
- b_5_27 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- b_5_28 → 0, an element of degree 5
- c_8_61 → c_1_12·c_1_26 + c_1_18 + c_1_0·c_1_27 + c_1_0·c_1_12·c_1_25
+ c_1_0·c_1_14·c_1_23 + c_1_02·c_1_26 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_03·c_1_25 + c_1_05·c_1_23 + c_1_06·c_1_22 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → c_1_2, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- c_2_8 → c_1_0·c_1_2 + c_1_02, an element of degree 2
- b_5_26 → c_1_25, an element of degree 5
- b_5_27 → 0, an element of degree 5
- b_5_28 → c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- c_8_61 → c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_27 + c_1_08, an element of degree 8
|