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Mod-5-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 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
This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(150,5); 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].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
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].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Restriction maps
Expressing the generators as elements of H*(SmallGroup(150,5); GF(5))
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a_3_0 →a_3_0 -
c_4_0 →c_4_0 -
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 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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