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Mod-3-Cohomology of AlternatingGroup(4), a group of order 12
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
General information on the group
- AlternatingGroup(4) is a group of order 12.
- The group order factors as
22 · 3 . - The group is defined by Group([(1,2,3),(2,3,4)]).
- 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(3,1); 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) − 1 + t - The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Benson 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 2:
-
a_1_0 , a nilpotent element of degree 1 -
c_2_0 , a Duflot element of degree 2
| 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 Benson test
- We proved completion in degree 2 using the Benson 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_2_0 , an element of degree 2
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, 1].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Restriction maps
Expressing the generators as elements of H*(SmallGroup(3,1); GF(3))
-
a_1_0 →a_1_0 -
c_2_0 →c_2_0
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1
-
a_1_0 →a_1_0 , an element of degree 1 -
c_2_0 →c_2_0 , an element of degree 2
| 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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