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Mod-13-Cohomology of U3(4), a group of order 62400
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
General information on the group
- U3(4) is a group of order 62400.
- The group order factors as
26 · 3 · 52 · 13 . - 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)(22,31)(23,32)(26,35)(28,37)(29,38)(30,40)(33,44)(34,42)(36,47)(39,51)(41,43)(45,55)(46,52)(48,57)(49,58)(50,59)(53,62)(54,60)(56,61)(63,64),(1,3,6)(2,4,8)(5,9,14)(7,11,17)(10,15,22)(12,18,26)(13,19,28)(16,23,33)(20,29,39)(21,30,41)(25,34,45)(27,36,48)(31,42,53)(32,43,54)(35,46,56)(37,49,57)(38,50,60)(40,52,47)(51,61,58)(55,63,65)(59,64,62)]).
- It is non-abelian.
- It has 13-Rank 1.
- The centre of a Sylow 13-subgroup has rank 1.
- Its Sylow 13-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(39,1); GF(13)).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) ( − 1 + t) · (1 − t + t2) · (1 + t + 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 6:
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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 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 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_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, 5].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Restriction maps
Expressing the generators as elements of H*(SmallGroup(39,1); GF(13))
-
a_5_0 →a_5_0 -
c_6_0 →c_6_0
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1
-
a_5_0 →c_2_02·a_1_0 , an element of degree 5 -
c_6_0 →c_2_03 , 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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