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Mod-3-Cohomology of AlternatingGroup(7), a group of order 2520
General information on the group
- AlternatingGroup(7) is a group of order 2520.
- The group order factors as 23 · 32 · 5 · 7.
- The group is defined by Group([(1,2,3,4,5,6,7),(5,6,7)]).
- 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(36,9); 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 − 2·t + 4·t2 − 4·t3 + 4·t4 − 4·t5 + 4·t6 − 2·t7 + t8 |
| ( − 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].
Ring generators
The cohomology ring has 8 minimal generators of maximal degree 8:
- a_2_0, a nilpotent element of degree 2
- a_3_0, a nilpotent element of degree 3
- a_3_1, a nilpotent element of degree 3
- c_4_0, a Duflot element of degree 4
- a_7_2, a nilpotent element of degree 7
- a_7_3, a nilpotent element of degree 7
- c_8_1, a Duflot element of degree 8
- c_8_2, a Duflot element of degree 8
Ring relations
There are 4 "obvious" relations:
a_3_02, a_3_12, a_7_22, a_7_32
Apart from that, there are 16 minimal relations of maximal degree 16:
- a_2_02
- a_2_0·a_3_0
- a_2_0·a_3_1
- a_3_0·a_3_1 + a_2_0·c_4_0
- a_2_0·a_7_2
- a_2_0·a_7_3
- a_3_0·a_7_2 + a_2_0·c_8_2
- a_3_0·a_7_3 + a_2_0·c_8_1
- a_3_1·a_7_2 − a_2_0·c_8_1
- a_3_1·a_7_3 + a_2_0·c_8_2
- c_8_2·a_3_0 − c_8_1·a_3_1 + c_4_0·a_7_3
- c_8_2·a_3_1 + c_8_1·a_3_0 − c_4_0·a_7_2
- a_7_2·a_7_3 + a_2_0·c_4_0·c_8_2
- c_8_2·a_7_2 + c_8_1·a_7_3 + c_4_0·c_8_1·a_3_0 − c_4_02·a_7_2
- c_8_2·a_7_3 − c_8_1·a_7_2 + c_4_0·c_8_1·a_3_1 − c_4_02·a_7_3
- c_8_22 + c_8_12 − c_4_02·c_8_2
Data used for the Hilbert-Poincaré test
- We proved completion in degree 16 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
- c_8_2, 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].
Restriction maps
- a_2_0 → a_2_0
- a_3_0 → a_3_0
- a_3_1 → a_3_1
- c_4_0 → c_4_0
- a_7_2 → a_7_2
- a_7_3 → a_7_3
- c_8_1 → c_8_1
- c_8_2 → c_8_2
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2
- a_2_0 → a_1_0·a_1_1, an element of degree 2
- a_3_0 → c_2_2·a_1_1 + c_2_1·a_1_0, an element of degree 3
- a_3_1 → c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
- c_4_0 → c_2_22 + c_2_12, an element of degree 4
- a_7_2 → c_2_1·c_2_22·a_1_1 − c_2_12·c_2_2·a_1_0, an element of degree 7
- a_7_3 → 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_1·c_2_23 − c_2_13·c_2_2, an element of degree 8
- c_8_2 → c_2_12·c_2_22, an element of degree 8
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