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Mod-5-Cohomology of AlternatingGroup(11), a group of order 19958400
General information on the group
- AlternatingGroup(11) is a group of order 19958400.
- The group order factors as 27 · 34 · 52 · 7 · 11.
- The group is defined by Group([(1,2,3,4,5,6,7,8,9,10,11),(9,10,11)]).
- It is non-abelian.
- It has 5-Rank 2.
- The centre of a Sylow 5-subgroup has rank 2.
- Its Sylow 5-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(400,206); GF(5)).
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 − 2·t3 + 2·t4 − 2·t5 + 4·t6 − 2·t7 + 2·t8 − 4·t9 + 2·t10 − 2·t11 + 4·t12 − 2·t13 + 2·t14 − 2·t15 + t16 − t17 + t18) |
| ( − 1 + t)2 · (1 + t2)2 · (1 + t4)2 · (1 + t8) |
- 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 16:
- a_6_0, a nilpotent element of degree 6
- a_7_0, a nilpotent element of degree 7
- a_7_1, a nilpotent element of degree 7
- c_8_0, a Duflot element of degree 8
- a_15_2, a nilpotent element of degree 15
- a_15_3, a nilpotent element of degree 15
- c_16_1, a Duflot element of degree 16
- c_16_2, a Duflot element of degree 16
Ring relations
There are 4 "obvious" relations:
a_7_02, a_7_12, a_15_22, a_15_32
Apart from that, there are 16 minimal relations of maximal degree 32:
- a_6_02
- a_6_0·a_7_0
- a_6_0·a_7_1
- a_7_0·a_7_1 + a_6_0·c_8_0
- a_6_0·a_15_2
- a_6_0·a_15_3
- a_7_0·a_15_2 − 2·a_6_0·c_16_2
- a_7_0·a_15_3 + a_6_0·c_16_1
- a_7_1·a_15_2 − a_6_0·c_16_1
- a_7_1·a_15_3 − 2·a_6_0·c_16_2
- c_16_2·a_7_0 − 2·c_16_1·a_7_1 + 2·c_8_0·a_15_3
- c_16_2·a_7_1 + 2·c_16_1·a_7_0 − 2·c_8_0·a_15_2
- a_15_2·a_15_3 + a_6_0·c_8_0·c_16_2
- c_16_2·a_15_2 + 2·c_16_1·a_15_3 − c_8_0·c_16_1·a_7_0 + c_8_02·a_15_2
- c_16_2·a_15_3 − 2·c_16_1·a_15_2 − c_8_0·c_16_1·a_7_1 + c_8_02·a_15_3
- c_16_22 − c_16_12 + c_8_02·c_16_2
Data used for the Hilbert-Poincaré test
- We proved completion in degree 32 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_8_0, an element of degree 8
- c_16_2, an element of degree 16
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 22].
Restriction maps
- a_6_0 → a_6_0
- a_7_0 → a_7_0
- a_7_1 → a_7_1
- c_8_0 → c_8_0
- a_15_2 → a_15_2
- a_15_3 → a_15_3
- c_16_1 → c_16_1
- c_16_2 → c_16_2
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2
- a_6_0 → c_2_1·c_2_2·a_1_0·a_1_1, an element of degree 6
- a_7_0 → c_2_23·a_1_1 + c_2_13·a_1_0, an element of degree 7
- 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_0 → c_2_24 + c_2_14, an element of degree 8
- a_15_2 → c_2_12·c_2_25·a_1_1 − c_2_15·c_2_22·a_1_0, an element of degree 15
- a_15_3 → c_2_13·c_2_24·a_1_0 + c_2_14·c_2_23·a_1_1, an element of degree 15
- c_16_1 → c_2_12·c_2_26 − c_2_16·c_2_22, an element of degree 16
- c_16_2 → c_2_14·c_2_24, an element of degree 16
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