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Mod-3-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 3-Rank 2.
- The centre of a Sylow 3-subgroup has rank 2.
- Its Sylow 3-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(144,186); GF(3)).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) · (1 − t + t2 − t3 + t4 − t5 + t6) ( − 1 + t)2 · (1 + t2)2 · (1 + t4) - 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].
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Ring generators
The cohomology ring has 4 minimal generators of maximal degree 8:
-
a_3_0 , a nilpotent element of degree 3 -
c_4_0 , a Duflot element of degree 4 -
a_7_1 , a nilpotent element of degree 7 -
c_8_1 , a Duflot element of degree 8
| 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 10 using the Hilbert-Poincaré criterion.
- However, the last relation was already found in degree 0 and the last generator in degree 8.
- The following is a filter regular homogeneous system of parameters:
c_4_0 , an element of degree 4c_8_1 , 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, -1, 10].
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Restriction maps
Expressing the generators as elements of H*(SmallGroup(144,186); GF(3))
-
a_3_0 →a_3_0 -
c_4_0 →c_4_0 -
a_7_1 →a_7_1 -
c_8_1 →c_8_1
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2
-
a_3_0 →c_2_2·a_1_1 + c_2_1·a_1_0 , an element of degree 3 -
c_4_0 →c_2_22 + c_2_12 , an element of degree 4 -
a_7_1 →c_2_1·c_2_22·a_1_0 + c_2_12·c_2_2·a_1_1 , an element of degree 7 -
c_8_1 →c_2_12·c_2_22 , 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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