Simon King
David J. Green
Cohomology
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Cohomology of group number 82 of order 128
General information on the group
- The group has 2 minimal generators and exponent 16.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 2.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is 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
t5 − t4 − t2 − 1 |
| (t + 1) · (t − 1)3 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 12 minimal generators of maximal degree 5:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- a_2_2, a nilpotent element of degree 2
- b_2_1, an element of degree 2
- a_3_2, a nilpotent element of degree 3
- a_3_3, a nilpotent element of degree 3
- b_3_1, an element of degree 3
- a_4_4, a nilpotent element of degree 4
- b_4_3, an element of degree 4
- c_4_5, a Duflot regular element of degree 4
- c_4_6, a Duflot regular element of degree 4
- b_5_8, an element of degree 5
Ring relations
There are 44 minimal relations of maximal degree 10:
- a_1_02
- a_1_0·a_1_1
- a_2_2·a_1_1
- a_2_2·a_1_0
- b_2_1·a_1_1
- b_2_1·a_1_0 + a_1_13
- a_2_22
- a_1_1·a_3_2
- a_1_0·a_3_2
- a_1_0·a_3_3
- a_1_1·b_3_1
- a_1_0·b_3_1
- a_2_2·a_3_3
- b_2_1·a_3_3 + a_2_2·a_3_2
- b_2_1·a_3_2 + a_2_2·b_3_1 + a_1_12·a_3_3
- a_4_4·a_1_1
- a_4_4·a_1_0 + a_2_2·a_3_2
- b_4_3·a_1_1 + a_2_2·a_3_2
- b_4_3·a_1_0 + a_1_12·a_3_3
- a_3_22
- a_3_2·a_3_3
- b_3_12 + b_2_13
- a_3_2·b_3_1 + a_2_2·b_2_12
- a_3_3·b_3_1
- a_3_32 + c_4_5·a_1_12
- a_2_2·a_4_4
- b_2_1·a_4_4 + a_2_2·b_4_3 + a_2_2·b_2_12
- a_1_1·b_5_8 + a_3_32
- a_1_0·b_5_8
- a_4_4·a_3_2
- a_4_4·a_3_3
- b_4_3·a_3_2 + a_4_4·b_3_1 + a_2_2·b_2_1·b_3_1 + c_4_5·a_1_13
- b_4_3·a_3_3
- a_4_4·b_3_1 + a_2_2·b_5_8
- b_4_3·b_3_1 + b_2_1·b_5_8 + b_2_12·b_3_1 + c_4_5·a_1_13
- a_4_42
- b_4_32 + b_2_12·c_4_5
- a_4_4·b_4_3 + a_2_2·b_2_1·b_4_3 + a_2_2·b_2_1·c_4_5
- b_3_1·b_5_8 + b_2_12·b_4_3 + b_2_14
- a_3_2·b_5_8 + a_2_2·b_2_1·b_4_3 + a_2_2·b_2_13
- a_3_3·b_5_8 + c_4_5·a_1_1·a_3_3
- b_4_3·b_5_8 + b_2_12·b_5_8 + b_2_13·b_3_1 + b_2_1·c_4_5·b_3_1 + a_2_2·c_4_6·a_3_2
+ c_4_5·a_1_12·a_3_3
- a_4_4·b_5_8 + a_2_2·b_2_12·b_3_1 + a_2_2·c_4_5·b_3_1 + a_2_2·c_4_5·a_3_2
- b_5_82 + b_2_15 + b_2_13·c_4_5 + c_4_52·a_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_4_5, a Duflot regular element of degree 4
- c_4_6, a Duflot regular element of degree 4
- b_2_1, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
- 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
- a_1_1 → 0, an element of degree 1
- a_2_2 → 0, an element of degree 2
- b_2_1 → 0, an element of degree 2
- a_3_2 → 0, an element of degree 3
- a_3_3 → 0, an element of degree 3
- b_3_1 → 0, an element of degree 3
- a_4_4 → 0, an element of degree 4
- b_4_3 → 0, an element of degree 4
- c_4_5 → c_1_14, an element of degree 4
- c_4_6 → c_1_14 + c_1_04, an element of degree 4
- b_5_8 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_2_2 → 0, an element of degree 2
- b_2_1 → c_1_22, an element of degree 2
- a_3_2 → 0, an element of degree 3
- a_3_3 → 0, an element of degree 3
- b_3_1 → c_1_23, an element of degree 3
- a_4_4 → 0, an element of degree 4
- b_4_3 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- c_4_5 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- c_4_6 → c_1_24 + c_1_1·c_1_23 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_5_8 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
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