Mathematics:
Cohomology
→Theory
→Implementation
Jena:
External links:
Mod-7-Cohomology of SymmetricGroup(8), a group of order 40320
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
General information on the group
- SymmetricGroup(8) is a group of order 40320.
- The group order factors as
27 · 32 · 5 · 7 . - The group is defined by Group([(1,2,3,4,5,6,7,8),(1,2)]).
- 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(42,1); 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 − t3 + t4 − t5 + t6 − t7 + t8 − t9 + t10) ( − 1 + t) · (1 − t + t2) · (1 + t2) · (1 + t + t2) · (1 − t2 + 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 12:
-
a_11_0 , a nilpotent element of degree 11 -
c_12_0 , a Duflot element of degree 12
| 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 12 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_12_0 , an element of degree 12
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, 11].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Restriction maps
Expressing the generators as elements of H*(SmallGroup(42,1); GF(7))
-
a_11_0 →a_11_0 -
c_12_0 →c_12_0
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1
-
a_11_0 →c_2_05·a_1_0 , an element of degree 11 -
c_12_0 →c_2_06 , an element of degree 12
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
|
Simon King Department of Mathematics and Computer Science Friedrich-Schiller-Universität Jena 07737 Jena GERMANY
|