Simon King
David J. Green
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Cohomology of group number 996 of order 128
General information on the group
- The group has 3 minimal generators and exponent 32.
- 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 1.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t6 + t5 + t2 + t + 1 |
| (t − 1)2 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-4,-2. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 6 minimal generators of maximal degree 8:
- a_1_1, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- b_1_2, an element of degree 1
- a_5_6, a nilpotent element of degree 5
- b_5_5, an element of degree 5
- c_8_10, a Duflot regular element of degree 8
Ring relations
There are 9 minimal relations of maximal degree 10:
- a_1_1·b_1_0
- b_1_0·b_1_22 + a_1_13
- a_1_13·b_1_22
- b_1_0·a_5_6
- a_1_1·b_5_5
- b_1_22·b_5_5 + a_1_12·a_5_6
- a_5_6·b_5_5
- a_5_62 + a_1_1·b_1_24·a_5_6 + a_1_12·b_1_23·a_5_6 + c_8_10·a_1_12
- b_5_52 + b_1_04·b_1_2·b_5_5 + c_8_10·b_1_02
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_8_10, a Duflot regular element of degree 8
- b_1_22 + b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, 4, 8].
- 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_1 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_5_6 → 0, an element of degree 5
- b_5_5 → 0, an element of degree 5
- c_8_10 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 2
- a_1_1 → 0, an element of degree 1
- b_1_0 → c_1_1, an element of degree 1
- b_1_2 → 0, an element of degree 1
- 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
- c_8_10 → 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_1 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- a_5_6 → 0, an element of degree 5
- b_5_5 → 0, an element of degree 5
- c_8_10 → c_1_04·c_1_14 + c_1_08, an element of degree 8
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