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Mod-2-Cohomology of group number 5 of order 60
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
General information on the group
- The group is also known as AlternatingGroup(5).
- The group order factors as
22 · 3 · 5 . - It is non-abelian.
- It has 2-Rank 2.
- The centre of a Sylow 2-subgroup has rank 2.
- Its Sylow 2-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.
Structure of the cohomology ring
This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(12,3); GF(2)).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 ( − 1 + t)2 · (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].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Ring generators
The cohomology ring has 3 minimal generators of maximal degree 3:
-
c_2_0 , a Duflot element of degree 2 -
c_3_0 , a Duflot element of degree 3 -
c_3_1 , a Duflot element of degree 3
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Ring relations
There is one minimal relation of degree 6:
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c_3_12 + c_3_0·c_3_1 + c_3_02 + c_2_03
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Data used for the Hilbert-Poincaré test
- We proved completion in degree 6 using the Hilbert-Poincaré criterion.
- 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_2_0 , an element of degree 2c_3_1 , an element of degree 3
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 3].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Restriction maps
Expressing the generators as elements of H*(SmallGroup(12,3); GF(2))
-
c_2_0 →c_2_0 -
c_3_0 →c_3_0 -
c_3_1 →c_3_1
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2
-
c_2_0 →c_1_12 + c_1_0·c_1_1 + c_1_02 , an element of degree 2 -
c_3_0 →c_1_13 + c_1_02·c_1_1 + c_1_03 , an element of degree 3 -
c_3_1 →c_1_0·c_1_12 + c_1_02·c_1_1 , an element of degree 3
| 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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