Simon King
David J. Green
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Cohomology of group number 893 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 3.
- Its center has rank 1.
- 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 1.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t6 − t5 − t4 + t3 − 1 |
| (t − 1)3 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-5,-3,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 10 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
- b_2_4, an element of degree 2
- a_4_5, a nilpotent element of degree 4
- a_5_6, a nilpotent element of degree 5
- a_5_8, a nilpotent element of degree 5
- b_5_9, an element of degree 5
- a_8_11, a nilpotent element of degree 8
- c_8_17, a Duflot regular element of degree 8
Ring relations
There are 29 minimal relations of maximal degree 16:
- a_1_02
- a_1_12 + a_1_0·a_1_1
- a_1_1·b_1_22 + b_2_4·a_1_0
- b_2_4·a_1_0·b_1_22 + b_2_42·a_1_0
- a_4_5·a_1_1
- a_4_5·a_1_0
- a_4_5·b_1_22 + a_1_0·a_5_6
- a_1_1·a_5_8 + a_1_1·a_5_6
- a_1_1·a_5_6 + a_1_0·a_5_8
- a_1_1·b_5_9 + b_2_4·a_4_5 + b_2_42·a_1_1·b_1_2 + a_1_1·a_5_6
- a_1_0·b_5_9 + b_2_42·a_1_0·b_1_2
- b_1_22·a_5_8 + b_2_4·a_5_6 + b_2_43·a_1_1
- a_4_52
- a_4_5·a_5_6
- a_4_5·a_5_8
- a_4_5·b_5_9 + b_2_42·a_4_5·b_1_2 + b_2_44·a_1_1 + b_2_44·a_1_0
- a_8_11·a_1_1 + b_2_4·a_1_0·b_1_2·a_5_6
- a_8_11·a_1_0 + b_2_4·a_1_0·b_1_2·a_5_6
- a_5_62 + a_1_0·b_1_24·a_5_6
- a_5_82 + a_5_6·a_5_8 + b_2_42·a_1_0·a_5_6
- b_5_92 + b_2_43·b_1_24 + b_2_44·b_1_22 + b_2_45
- a_5_6·a_5_8 + c_8_17·a_1_0·a_1_1
- a_5_6·b_5_9 + a_8_11·b_1_22 + b_2_4·b_1_23·a_5_6 + b_2_42·b_1_2·a_5_6
+ b_2_43·a_4_5 + b_2_44·a_1_0·b_1_2 + b_2_42·a_1_0·a_5_6
- a_5_8·b_5_9 + b_2_4·a_8_11 + b_2_42·b_1_2·a_5_8 + b_2_42·b_1_2·a_5_6 + b_2_43·a_4_5
+ b_2_44·a_1_0·b_1_2 + a_5_6·a_5_8 + b_2_42·a_1_0·a_5_6
- a_4_5·a_8_11
- a_8_11·a_5_6
- a_8_11·a_5_8 + b_2_43·a_1_0·b_1_2·a_5_6
- a_8_11·b_5_9 + b_2_4·a_8_11·b_1_23 + b_2_42·b_1_24·a_5_6 + b_2_42·a_8_11·b_1_2
+ b_2_43·b_1_22·a_5_6 + b_2_44·a_5_8 + b_2_44·a_4_5·b_1_2 + b_2_46·a_1_1 + b_2_46·a_1_0 + b_2_43·a_1_0·b_1_2·a_5_6
- a_8_112 + b_2_45·a_1_0·a_5_6
Data used for Benson′s test
- Benson′s completion test succeeded in degree 16.
- 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_17, a Duflot regular element of degree 8
- b_1_24 + b_2_4·b_1_22 + b_2_42, an element of degree 4
- b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, 3, 9, 11].
- 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 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
- b_2_4 → 0, an element of degree 2
- a_4_5 → 0, an element of degree 4
- a_5_6 → 0, an element of degree 5
- a_5_8 → 0, an element of degree 5
- b_5_9 → 0, an element of degree 5
- a_8_11 → 0, an element of degree 8
- c_8_17 → c_1_08, an element of degree 8
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
- b_1_2 → c_1_1, an element of degree 1
- b_2_4 → c_1_22, an element of degree 2
- a_4_5 → 0, an element of degree 4
- a_5_6 → 0, an element of degree 5
- a_5_8 → 0, an element of degree 5
- b_5_9 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
- a_8_11 → 0, an element of degree 8
- c_8_17 → c_1_1·c_1_27 + c_1_15·c_1_23 + 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_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
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