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Mod-2-Cohomology of AlternatingGroup(10), a group of order 1814400
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
General information on the group
- AlternatingGroup(10) is a group of order 1814400.
- The group order factors as
27 · 34 · 52 · 7 . - The group is defined by Group([(1,2,3,4,5,6,7,8,9),(8,9,10)]).
- It is non-abelian.
- It has 2-Rank 4.
- The centre of a Sylow 2-subgroup has rank 1.
- Its Sylow 2-subgroup has 5 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3, 4, 4 and 4, respectively.
Structure of the cohomology ring
The computation was based on 15 stability conditions for H*(SmallGroup(384,5602); GF(2)).General information
- The cohomology ring is of dimension 4 and depth 3.
- The depth exceeds the Duflot bound, which is 1.
- The Poincaré series is
1 + 2·t2 + t3 + 3·t4 + 3·t5 + 5·t6 + 5·t7 + 7·t8 + 6·t9 + 6·t10 + 6·t11 + 4·t12 + 4·t13 + 3·t14 + 3·t15 + t16 + 2·t17 + t19 (1 + t) · ( − 1 + t)4 · (1 − t + t2) · (1 + t2) · (1 + t + t2)2 · (1 + t + t2 + t3 + t4) · (1 + t + t2 + t3 + t4 + t5 + t6) - The a-invariants are -∞,-∞,-∞,-10,-4. 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, -3, -4, -4].
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Ring generators
The cohomology ring has 8 minimal generators of maximal degree 7:
-
b_2_0 , an element of degree 2 -
b_3_1 , an element of degree 3 -
b_3_0 , an element of degree 3 -
c_4_0 , a Duflot element of degree 4 -
b_5_2 , an element of degree 5 -
b_5_0 , an element of degree 5 -
b_6_0 , an element of degree 6 -
b_7_6 , an element of degree 7
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Ring relations
There are 10 minimal relations of maximal degree 12:
-
b_3_1·b_5_0 + b_3_0·b_5_2 + b_2_0·b_3_0·b_3_1 -
b_2_0·b_7_6 -
b_6_0·b_3_0 -
b_3_0·b_7_6 -
b_3_1·b_7_6 -
b_5_0·b_5_2 + b_2_02·b_3_0·b_3_1 + c_4_0·b_3_0·b_3_1 -
b_5_22 + b_2_0·b_3_1·b_5_2 + b_2_02·b_3_12 + b_2_02·b_6_0 + c_4_0·b_3_12 -
b_6_0·b_5_0 -
b_5_0·b_7_6 -
b_5_2·b_7_6
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Data used for the Hilbert-Poincaré test
- We proved completion in degree 29 using the Hilbert-Poincaré criterion.
- However, the last relation was already found in degree 12 and the last generator in degree 7.
- The following is a filter regular homogeneous system of parameters:
b_3_1·b_5_2 + b_3_0·b_5_0 + b_2_0·b_3_12 + b_2_0·b_3_0·b_3_1 + b_2_0·b_3_02 + c_4_02 , an element of degree 8b_3_14 + b_3_0·b_3_13 + b_3_02·b_3_12 + b_3_03·b_3_1 + b_3_04 + b_6_0·b_3_12
+ b_6_02 + b_2_0·b_5_02 + b_2_02·b_3_1·b_5_2 + b_2_03·b_3_02 + b_2_03·b_6_0
+ b_2_06 + c_4_0·b_3_1·b_5_2 + c_4_0·b_3_0·b_5_0 + b_2_0·c_4_0·b_3_0·b_3_1
+ b_2_0·c_4_0·b_3_02 + b_2_02·c_4_02 , an element of degree 12b_7_62 + b_3_13·b_5_2 + b_3_0·b_3_12·b_5_2 + b_3_03·b_5_2 + b_3_03·b_5_0
+ b_6_0·b_3_1·b_5_2 + b_2_0·b_3_14 + b_2_0·b_3_0·b_3_13 + b_2_0·b_3_04
+ b_2_03·b_3_1·b_5_2 + b_2_03·b_3_0·b_5_0 + b_2_04·b_3_0·b_3_1 + c_4_0·b_5_02
+ b_2_0·c_4_0·b_3_0·b_5_0 + b_2_02·c_4_0·b_3_12 + b_2_02·c_4_0·b_3_0·b_3_1
+ b_2_02·c_4_0·b_6_0 + c_4_02·b_3_02 , an element of degree 14b_5_03 + b_3_0·b_3_14 + b_3_02·b_3_13 + b_3_03·b_3_12 + b_3_04·b_3_1
+ b_6_0·b_3_13 + b_2_02·b_3_0·b_3_1·b_5_2 + b_2_02·b_3_02·b_5_2
+ b_2_02·b_6_0·b_5_2 + b_2_03·b_3_0·b_3_12 + b_2_03·b_3_03 + b_2_03·b_6_0·b_3_1
+ c_4_0·b_3_02·b_5_0 + b_2_0·c_4_0·b_3_03 , an element of degree 15
- A Duflot regular sequence is given by
c_4_0 . - The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 24, 45].
- Modifying the above filter regular HSOP, we obtained the following parameters:
c_4_0 , an element of degree 4b_3_12 + b_3_0·b_3_1 + b_3_02 + b_6_0 + b_2_03 , an element of degree 6b_7_6 + b_2_02·b_3_1 + b_2_02·b_3_0 , an element of degree 7b_5_03 + b_3_0·b_3_14 + b_3_02·b_3_13 + b_3_03·b_3_12 + b_3_04·b_3_1
+ b_6_0·b_3_13 + b_2_02·b_3_0·b_3_1·b_5_2 + b_2_02·b_3_02·b_5_2
+ b_2_02·b_6_0·b_5_2 + b_2_03·b_3_0·b_3_12 + b_2_03·b_3_03 + b_2_03·b_6_0·b_3_1
+ c_4_0·b_3_02·b_5_0 + b_2_0·c_4_0·b_3_03 , an element of degree 15
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Restriction maps
Expressing the generators as elements of H*(SmallGroup(384,5602); GF(2))
-
b_2_0 →b_1_0·b_1_1 + b_1_02 + b_2_4 -
b_3_1 →b_3_0 + b_1_0·b_1_12 + b_1_02·b_1_1 -
b_3_0 →b_3_1 + b_2_4·b_1_1 -
c_4_0 →b_1_14 + b_1_0·b_3_1 + b_1_0·b_3_0 + b_1_0·b_1_13 + b_1_04 + b_2_4·b_1_12
+ b_2_4·b_1_0·b_1_1 + b_2_4·b_1_02 + b_2_42 + b_2_3·b_2_4 + b_2_32 + c_4_15 -
b_5_2 →b_1_0·b_1_14 + b_1_02·b_3_0 + b_1_04·b_1_1 + b_2_4·b_3_0 + b_2_3·b_3_0
+ b_2_3·b_2_4·b_1_0 + c_4_15·b_1_0 -
b_5_0 →b_1_12·b_3_1 + b_2_4·b_3_1 + b_2_4·b_1_13 + b_2_42·b_1_1 + c_4_15·b_1_1 -
b_6_0 →b_3_92 + b_2_3·c_4_15 -
b_7_6 →c_4_15·b_3_9
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1
-
b_2_0 →0 , an element of degree 2 -
b_3_1 →0 , an element of degree 3 -
b_3_0 →0 , an element of degree 3 -
c_4_0 →c_1_04 , an element of degree 4 -
b_5_2 →0 , an element of degree 5 -
b_5_0 →0 , an element of degree 5 -
b_6_0 →0 , an element of degree 6 -
b_7_6 →0 , an element of degree 7
Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup
-
b_2_0 →c_1_1·c_1_2 + c_1_12 , an element of degree 2 -
b_3_1 →c_1_1·c_1_22 + c_1_12·c_1_2 , an element of degree 3 -
b_3_0 →0 , an element of degree 3 -
c_4_0 →c_1_24 + c_1_1·c_1_23 + c_1_14 + c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2
+ c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2 + c_1_02·c_1_12 + c_1_04 , an element of degree 4 -
b_5_2 →c_1_1·c_1_24 + c_1_14·c_1_2 , an element of degree 5 -
b_5_0 →0 , an element of degree 5 -
b_6_0 →c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_24
+ c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_22 + c_1_04·c_1_22 , an element of degree 6 -
b_7_6 →0 , an element of degree 7
Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup
-
b_2_0 →0 , an element of degree 2 -
b_3_1 →0 , an element of degree 3 -
b_3_0 →0 , an element of degree 3 -
c_4_0 →c_1_24 + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2
+ c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2 + c_1_02·c_1_12 + c_1_04 , an element of degree 4 -
b_5_2 →0 , an element of degree 5 -
b_5_0 →0 , an element of degree 5 -
b_6_0 →c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_14·c_1_2
+ c_1_02·c_1_24 + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_14 + c_1_04·c_1_22
+ c_1_04·c_1_1·c_1_2 + c_1_04·c_1_12 , an element of degree 6 -
b_7_6 →c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
+ c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2 , an element of degree 7
Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup
-
b_2_0 →c_1_32 + c_1_2·c_1_3 + c_1_1·c_1_3 + c_1_12 + c_1_0·c_1_3 + c_1_0·c_1_2 , an element of degree 2 -
b_3_1 →c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32
+ c_1_02·c_1_3 , an element of degree 3 -
b_3_0 →c_1_0·c_1_22 + c_1_02·c_1_2 , an element of degree 3 -
c_4_0 →c_1_34 + c_1_23·c_1_3 + c_1_24 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32
+ c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_23
+ c_1_0·c_1_1·c_1_2·c_1_3 + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_22
+ c_1_02·c_1_1·c_1_3 + c_1_02·c_1_12 + c_1_03·c_1_3 + c_1_03·c_1_2 + c_1_04 , an element of degree 4 -
b_5_2 →c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_1·c_1_34 + c_1_14·c_1_3 + c_1_0·c_1_34
+ c_1_04·c_1_3 , an element of degree 5 -
b_5_0 →c_1_0·c_1_24 + c_1_04·c_1_2 , an element of degree 5 -
b_6_0 →0 , an element of degree 6 -
b_7_6 →0 , an element of degree 7
Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup
-
b_2_0 →c_1_32 + c_1_2·c_1_3 + c_1_1·c_1_3 + c_1_12 + c_1_0·c_1_2 , an element of degree 2 -
b_3_1 →0 , an element of degree 3 -
b_3_0 →c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_22
+ c_1_02·c_1_2 , an element of degree 3 -
c_4_0 →c_1_34 + c_1_23·c_1_3 + c_1_24 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32
+ c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_23 + c_1_0·c_1_1·c_1_32
+ c_1_0·c_1_1·c_1_2·c_1_3 + c_1_0·c_1_12·c_1_3 + c_1_0·c_1_12·c_1_2
+ c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_3
+ c_1_02·c_1_12 + c_1_03·c_1_2 + c_1_04 , an element of degree 4 -
b_5_2 →0 , an element of degree 5 -
b_5_0 →c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_1·c_1_34 + c_1_14·c_1_3 + c_1_0·c_1_24
+ c_1_04·c_1_2 , an element of degree 5 -
b_6_0 →0 , an element of degree 6 -
b_7_6 →0 , an element of degree 7
Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup
-
b_2_0 →c_1_22 + c_1_1·c_1_3 + c_1_1·c_1_2 + c_1_12 + c_1_0·c_1_2 , an element of degree 2 -
b_3_1 →c_1_1·c_1_32 + c_1_1·c_1_22 + c_1_12·c_1_3 + c_1_12·c_1_2 + c_1_0·c_1_22
+ c_1_02·c_1_2 , an element of degree 3 -
b_3_0 →0 , an element of degree 3 -
c_4_0 →c_1_34 + c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32
+ c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3
+ c_1_0·c_1_1·c_1_32 + c_1_0·c_1_1·c_1_2·c_1_3 + c_1_0·c_1_12·c_1_3
+ c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_3 + c_1_02·c_1_1·c_1_2
+ c_1_02·c_1_12 + c_1_03·c_1_2 + c_1_04 , an element of degree 4 -
b_5_2 →c_1_1·c_1_34 + c_1_1·c_1_24 + c_1_14·c_1_3 + c_1_14·c_1_2 + c_1_0·c_1_24
+ c_1_04·c_1_2 , an element of degree 5 -
b_5_0 →0 , an element of degree 5 -
b_6_0 →c_1_0·c_1_1·c_1_34 + c_1_0·c_1_1·c_1_22·c_1_32 + c_1_0·c_1_12·c_1_33
+ c_1_0·c_1_12·c_1_2·c_1_32 + c_1_02·c_1_34 + c_1_02·c_1_22·c_1_32
+ c_1_02·c_1_1·c_1_33 + c_1_02·c_1_1·c_1_22·c_1_3 + c_1_02·c_1_12·c_1_32
+ c_1_02·c_1_12·c_1_2·c_1_3 + c_1_03·c_1_2·c_1_32 + c_1_03·c_1_22·c_1_3
+ c_1_04·c_1_32 + c_1_04·c_1_2·c_1_3 , an element of degree 6 -
b_7_6 →0 , an element of degree 7
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Simon King Department of Mathematics and Computer Science Friedrich-Schiller-Universität Jena 07737 Jena GERMANY
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