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Mod-5-Cohomology of SymplecticGroup(4,3), a group of order 51840
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
General information on the group
- SymplecticGroup(4,3) is a group of order 51840.
- The group order factors as
27 · 34 · 5 . - The group is defined by Group([(1,2)(3,5)(4,7)(6,10)(8,12)(9,13)(11,16)(14,20)(15,21)(17,24)(18,25)(19,27)(23,31)(28,32)(29,37)(30,39)(33,38)(34,43)(35,45)(41,49)(46,54)(47,51)(48,57)(50,53)(52,61)(55,56)(58,67)(59,60)(62,70)(65,71)(66,72)(68,74)(69,75)(73,76)(77,79)(78,80),(1,3,6)(2,4,8)(5,9,14,12,18,26,34,44,25)(7,11,17,10,15,22,30,40,21)(13,19,28,36,47,56,20,29,38)(16,23,32,42,51,60,24,33,37)(27,35,46,55,65,67,73,57,66)(31,41,50,59,68,70,76,61,69)(39,48,58)(43,52,62)(45,53,63)(49,54,64)(71,77,75)(72,74,78)]).
- It is non-abelian.
- It has 5-Rank 1.
- The centre of a Sylow 5-subgroup has rank 1.
- Its Sylow 5-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.
Structure of the cohomology ring
This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(40,3); GF(5)).General information
- The cohomology ring is of dimension 1 and depth 1.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1)·(1 − t + t2 − t3 + t4 − t5 + t6) ( − 1 + t) · (1 + t2) · (1 + t4) - The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
- The filter degree type of any filter regular HSOP is [-1, -1].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Ring generators
The cohomology ring has 2 minimal generators of maximal degree 8:
-
a_7_0 , a nilpotent element of degree 7 -
c_8_0 , a Duflot element of degree 8
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Ring relations
There is one "obvious" relation:
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.
- 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_0 , an element of degree 8
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, 7].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Restriction maps
Expressing the generators as elements of H*(SmallGroup(40,3); GF(5))
-
a_7_0 →a_7_0 -
c_8_0 →c_8_0
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1
-
a_7_0 →c_2_03·a_1_0 , an element of degree 7 -
c_8_0 →c_2_04 , an element of degree 8
| 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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