Mod-3-Cohomology of SymmetricGroup(4), a group of order 24
General information on the group
- SymmetricGroup(4) is a group of order 24.
- The group order factors as 23 · 3.
- The group is defined by Group([(1,2,3,4),(1,2)]).
- It is non-abelian.
- It has 3-Rank 1.
- The centre of a Sylow 3-subgroup has rank 1.
- Its Sylow 3-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(6,1); GF(3)).
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].
Ring generators
The cohomology ring has 2 minimal generators of maximal degree 4:
- a_3_0, a nilpotent element of degree 3
- c_4_0, a Duflot element of degree 4
Ring relations
There is one "obvious" relation:
a_3_02
Apart from that, there are no relations.
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].
Restriction maps
- 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
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