Simon King
David J. Green
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Cohomology of group number 1735 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 5.
- Its center has rank 2.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3 and 5, respectively.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 5 and depth 3.
- The depth exceeds the Duflot bound, which is 2.
- The Poincaré series is
( − 1) · (t2 − t + 1) |
| (t − 1)5 · (t2 + 1) |
- The a-invariants are -∞,-∞,-∞,-5,-5,-5. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 10 minimal generators of maximal degree 4:
- 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
- b_2_8, an element of degree 2
- c_2_9, a Duflot regular element of degree 2
- b_3_19, an element of degree 3
- b_3_20, an element of degree 3
- b_4_37, an element of degree 4
- c_4_39, a Duflot regular element of degree 4
Ring relations
There are 18 minimal relations of maximal degree 8:
- b_1_0·b_1_1
- b_1_0·b_1_2
- b_1_0·b_1_32
- b_2_8·b_1_0
- b_2_8·b_1_1·b_1_2 + b_2_82 + c_2_9·b_1_12
- b_1_0·b_3_19
- b_1_0·b_3_20
- b_1_2·b_3_19 + b_1_1·b_3_20 + b_2_8·b_1_32
- b_4_37·b_1_2 + b_2_8·b_3_20 + b_2_8·b_1_2·b_1_32 + c_2_9·b_1_1·b_1_32
- b_4_37·b_1_0
- b_4_37·b_1_1 + b_2_8·b_3_19
- b_3_202 + b_1_2·b_1_32·b_3_20 + c_4_39·b_1_22 + c_2_9·b_1_34
- b_3_192 + b_1_1·b_1_32·b_3_19 + c_4_39·b_1_12
- b_2_8·b_1_1·b_3_20 + b_2_8·b_4_37 + b_2_82·b_1_32 + c_2_9·b_1_1·b_3_19
- b_3_19·b_3_20 + b_1_1·b_1_32·b_3_20 + b_4_37·b_1_32 + b_2_8·b_1_34
+ c_4_39·b_1_1·b_1_2
- b_4_37·b_3_19 + b_2_8·b_1_32·b_3_19 + b_2_8·c_4_39·b_1_1
- b_4_37·b_3_20 + c_2_9·b_1_32·b_3_19 + b_2_8·c_4_39·b_1_2
- b_4_372 + b_2_8·b_4_37·b_1_32 + b_2_82·c_4_39
Data used for Benson′s test
- Benson′s completion test succeeded in degree 13.
- However, the last relation was already found in degree 8 and the last generator in degree 4.
- The following is a filter regular homogeneous system of parameters:
- c_2_9, a Duflot regular element of degree 2
- c_4_39, a Duflot regular element of degree 4
- b_1_34 + b_1_22·b_1_32 + b_1_24 + b_1_1·b_1_2·b_1_32 + b_1_1·b_1_22·b_1_3
+ b_1_12·b_1_32 + b_1_12·b_1_2·b_1_3 + b_1_12·b_1_22 + b_1_14 + b_1_04, an element of degree 4
- b_1_22·b_1_34 + b_1_24·b_1_32 + b_1_1·b_1_2·b_1_34 + b_1_1·b_1_24·b_1_3
+ b_1_12·b_1_34 + b_1_12·b_1_22·b_1_32 + b_1_12·b_1_24 + b_1_14·b_1_32 + b_1_14·b_1_2·b_1_3 + b_1_14·b_1_22, an element of degree 6
- b_1_32, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 11, 13].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].
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
- b_2_8 → 0, an element of degree 2
- c_2_9 → c_1_02, an element of degree 2
- b_3_19 → 0, an element of degree 3
- b_3_20 → 0, an element of degree 3
- b_4_37 → 0, an element of degree 4
- c_4_39 → c_1_14, an element of degree 4
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 → 0, an element of degree 1
- b_2_8 → 0, an element of degree 2
- c_2_9 → c_1_0·c_1_2 + c_1_02, an element of degree 2
- b_3_19 → 0, an element of degree 3
- b_3_20 → 0, an element of degree 3
- b_4_37 → 0, an element of degree 4
- c_4_39 → c_1_12·c_1_22 + c_1_14, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 5
- 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_3, an element of degree 1
- b_1_3 → c_1_4, an element of degree 1
- b_2_8 → c_1_0·c_1_2, an element of degree 2
- c_2_9 → c_1_0·c_1_3 + c_1_02, an element of degree 2
- b_3_19 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- b_3_20 → c_1_1·c_1_2·c_1_3 + c_1_12·c_1_3 + c_1_0·c_1_42, an element of degree 3
- b_4_37 → c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2, an element of degree 4
- c_4_39 → c_1_1·c_1_2·c_1_42 + c_1_12·c_1_42 + c_1_12·c_1_22 + c_1_14, an element of degree 4
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