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Mod-3-Cohomology of MathieuGroup(23), a group of order 10200960
General information on the group
- MathieuGroup(23) is a group of order 10200960.
- The group order factors as 27 · 32 · 5 · 7 · 11 · 23.
- The group is defined by Group([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23),(3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16)]).
- 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(144,182); 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 + 3·t2 − 4·t3 + 4·t4 − 4·t5 + 4·t6 − 3·t7 + 3·t8 − 3·t9 + 4·t10 − 4·t11 + 4·t12 − 4·t13 + 3·t14 − 2·t15 + t16 |
| ( − 1 + t)2 · (1 + t2)2 · (1 + t4) · (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_7_0, a nilpotent element of degree 7
- c_8_0, a Duflot element of degree 8
- a_10_0, a nilpotent element of degree 10
- a_11_0, a nilpotent element of degree 11
- a_11_1, a nilpotent element of degree 11
- c_12_0, a Duflot element of degree 12
- a_15_1, a nilpotent element of degree 15
- c_16_1, a Duflot element of degree 16
Ring relations
There are 4 "obvious" relations:
a_7_02, a_11_02, a_11_12, a_15_12
Apart from that, there are 16 minimal relations of maximal degree 27:
- a_10_0·a_7_0
- a_7_0·a_11_0 − c_8_0·a_10_0
- a_7_0·a_11_1 + c_8_0·a_10_0
- c_12_0·a_7_0 − c_8_0·a_11_1 − c_8_0·a_11_0
- a_10_02
- a_10_0·a_11_0
- a_10_0·a_11_1
- a_7_0·a_15_1 + a_10_0·c_12_0
- a_11_0·a_11_1 + a_10_0·c_12_0
- c_12_0·a_11_0 + c_8_0·a_15_1 − c_8_02·a_7_0
- c_16_1·a_7_0 + c_12_0·a_11_1 − c_8_0·a_15_1
- c_12_02 + c_8_0·c_16_1 − c_8_03
- a_10_0·a_15_1
- a_11_0·a_15_1 + c_8_02·a_10_0
- a_11_1·a_15_1 − a_10_0·c_16_1
- c_16_1·a_11_0 − c_12_0·a_15_1 + c_8_02·a_11_1
Data used for the Hilbert-Poincaré test
- We proved completion in degree 27 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_1, 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_7_0 → a_7_0
- c_8_0 → c_8_0
- a_10_0 → a_10_0
- a_11_0 → a_11_0
- a_11_1 → a_11_1
- c_12_0 → c_12_0
- a_15_1 → a_15_1
- c_16_1 → c_16_1
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2
- a_7_0 → c_2_23·a_1_1 + c_2_1·c_2_22·a_1_0 + c_2_12·c_2_2·a_1_1 + c_2_13·a_1_0, an element of degree 7
- c_8_0 → c_2_24 − c_2_12·c_2_22 + c_2_14, an element of degree 8
- a_10_0 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
- a_11_0 → c_2_25·a_1_1 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0 + c_2_15·a_1_0, an element of degree 11
- a_11_1 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
+ c_2_14·c_2_2·a_1_1, an element of degree 11
- c_12_0 → c_2_26 + c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12
- a_15_1 → c_2_1·c_2_26·a_1_0 − c_2_13·c_2_24·a_1_0 − c_2_14·c_2_23·a_1_1
+ c_2_16·c_2_2·a_1_1, an element of degree 15
- c_16_1 → c_2_12·c_2_26 + c_2_14·c_2_24 + c_2_16·c_2_22, an element of degree 16
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