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Mod-7-Cohomology of SuzukiGroup(8), a group of order 29120
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
General information on the group
- SuzukiGroup(8) is a group of order 29120.
- The group order factors as
26 · 5 · 7 · 13 . - The group is defined by Group([(2,3,4,6)(5,8,11,16)(7,10,14,19)(9,13,18,24)(12,17,22,30)(15,20,27,34)(21,29,37,46)(23,32,41,25)(26,33,42,52)(28,36,44,54)(31,40,50,59)(35,38,48,57)(39,49,58,51)(43,53,61,65)(45,55,63,64)(47,56,60,62),(1,2)(3,5)(4,7)(6,9)(8,12)(10,15)(11,13)(16,21)(17,23)(18,25)(19,26)(20,28)(22,31)(24,32)(27,35)(29,38)(30,39)(33,42)(34,43)(36,45)(37,47)(40,51)(41,49)(44,54)(46,55)(48,52)(50,60)(53,62)(56,64)(57,65)(58,59)(61,63)]).
- It is non-abelian.
- It has 7-Rank 1.
- The centre of a Sylow 7-subgroup has rank 1.
- Its Sylow 7-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(14,1); GF(7)).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) ( − 1 + t) · (1 + t2) - 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 4:
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a_3_0 , a nilpotent element of degree 3 -
c_4_0 , a Duflot element of degree 4
| 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 4 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_4_0 , an element of degree 4
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, 3].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Restriction maps
Expressing the generators as elements of H*(SmallGroup(14,1); GF(7))
-
a_3_0 →a_3_0 -
c_4_0 →c_4_0
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1
-
a_3_0 →c_2_0·a_1_0 , an element of degree 3 -
c_4_0 →c_2_02 , an element of degree 4
| 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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