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Mod-5-Cohomology of group number 5 of order 150
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
General information on the group
- The group order factors as
2 · 3 · 52 . - It is non-abelian.
- It has 5-Rank 2.
- The centre of a Sylow 5-subgroup has rank 2.
- Its Sylow 5-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.
Structure of the cohomology ring
The computation was based on 5 stability conditions for H*(SmallGroup(25,2); GF(5)).General information
- The cohomology ring is of dimension 2 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
1 − t + t2 − t3 + t4 ( − 1 + t)2 · (1 + t2) · (1 + t + t2) - The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
- The filter degree type of any filter regular HSOP is [-1, -2, -2].
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Ring generators
The cohomology ring has 4 minimal generators of maximal degree 6:
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a_3_0 , a nilpotent element of degree 3 -
c_4_0 , a Duflot element of degree 4 -
a_5_0 , a nilpotent element of degree 5 -
c_6_0 , a Duflot element of degree 6
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Ring relations
There are 2 "obvious" relations:
Apart from that, there are no relations.
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Data used for the Hilbert-Poincaré test
- We proved completion in degree 8 using the Hilbert-Poincaré criterion.
- However, the last relation was already found in degree 0 and the last generator in degree 6.
- The following is a filter regular homogeneous system of parameters:
c_4_0 , an element of degree 4c_6_0 , an element of degree 6
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 8].
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Restriction maps
Expressing the generators as elements of H*(SmallGroup(25,2); GF(5))
-
a_3_0 →c_2_2·a_1_1 + 2·c_2_2·a_1_0 + 2·c_2_1·a_1_1 + c_2_1·a_1_0 -
c_4_0 →c_2_22 − c_2_1·c_2_2 + c_2_12 -
a_5_0 →c_2_22·a_1_1 + 2·c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1 − c_2_1·c_2_2·a_1_0
+ 2·c_2_12·a_1_1 + c_2_12·a_1_0 -
c_6_0 →c_2_23 + c_2_1·c_2_22 + c_2_12·c_2_2 + c_2_13
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2
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a_3_0 →c_2_2·a_1_1 + 2·c_2_2·a_1_0 + 2·c_2_1·a_1_1 + c_2_1·a_1_0 , an element of degree 3 -
c_4_0 →c_2_22 − c_2_1·c_2_2 + c_2_12 , an element of degree 4 -
a_5_0 →c_2_22·a_1_1 + 2·c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1 − c_2_1·c_2_2·a_1_0
+ 2·c_2_12·a_1_1 + c_2_12·a_1_0 , an element of degree 5 -
c_6_0 →c_2_23 + c_2_1·c_2_22 + c_2_12·c_2_2 + c_2_13 , an element of degree 6
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
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Simon King Department of Mathematics and Computer Science Friedrich-Schiller-Universität Jena 07737 Jena GERMANY
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