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Mod-5-Cohomology of Sym10, a group of order 3628800

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• Sym10 is a group of order 3628800.
• The group order factors as 2⁸ · 3⁴ · 5² · 7.
• The group is defined by Group([(1,2,3,4,5,6,7,8,9,10),(1,2)]).
• 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

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  −  t3  +  t4) · (1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6) · (1  +  t  −  t3  −  t4  −  t5  +  t7  +  t8)

  ( − 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].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 4 minimal generators of maximal degree 16:

1. a_7_0, a nilpotent element of degree 7
2. c_8_0, a Duflot element of degree 8
3. a_15_1, a nilpotent element of degree 15
4. c_16_1, a Duflot element of degree 16

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 2 "obvious" relations:
   a_7_0², a_15_1²

Apart from that, there are no relations.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 22 using the Hilbert-Poincaré criterion.
• However, the last relation was already found in degree 0 and the last generator in degree 16.
• The following is a filter regular homogeneous system of parameters:
  1. c_8_0, an element of degree 8
  2. 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].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(800,1191); GF(5))

1. a_7_0 → a_7_0
2. c_8_0 → c_8_0
3. a_15_1 → a_15_1
4. 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

1. a_7_0 → c_2_2³·a_1_1 + c_2_1³·a_1_0, an element of degree 7
2. c_8_0 → c_2_2⁴ + c_2_1⁴, an element of degree 8
3. a_15_1 → c_2_1³·c_2_2⁴·a_1_0 + c_2_1⁴·c_2_2³·a_1_1, an element of degree 15
4. c_16_1 → c_2_1⁴·c_2_2⁴, an element of degree 16

About the group

Ring generators

Ring relations

Completion information

Restriction maps