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Mod-2-Cohomology of AlternatingGroup(6), a group of order 360

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• AlternatingGroup(6) is a group of order 360.
• The group order factors as 2³ · 3² · 5.
• The group is defined by Group([(1,2,3,4,5),(4,5,6)]).
• It is non-abelian.
• It has 2-Rank 2.
• The centre of a Sylow 2-subgroup has rank 1.
• Its Sylow 2-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 2.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  1  −  t  +  t2

  ( − 1  +  t)2 · (1  +  t  +  t2)

• 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 3 minimal generators of maximal degree 3:

1. c_2_0, a Duflot element of degree 2
2. b_3_1, an element of degree 3
3. b_3_0, an element of degree 3

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There is one minimal relation of degree 6:

1. b_3_0·b_3_1

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 6 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:
  1. c_2_0, an element of degree 2
  2. b_3_1 + b_3_0, an element of degree 3
• A Duflot regular sequence is given by c_2_0.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 3].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(D8; GF(2))

1. c_2_0 → b_1_1² + b_1_0² + c_2_2
2. b_3_1 → c_2_2·b_1_0
3. b_3_0 → c_2_2·b_1_1

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

1. c_2_0 → c_1_0², an element of degree 2
2. b_3_1 → 0, an element of degree 3
3. b_3_0 → 0, an element of degree 3

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

1. c_2_0 → c_1_1² + c_1_0·c_1_1 + c_1_0², an element of degree 2
2. b_3_1 → c_1_0·c_1_1² + c_1_0²·c_1_1, an element of degree 3
3. b_3_0 → 0, an element of degree 3

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

1. c_2_0 → c_1_1² + c_1_0·c_1_1 + c_1_0², an element of degree 2
2. b_3_1 → 0, an element of degree 3
3. b_3_0 → c_1_0·c_1_1² + c_1_0²·c_1_1, an element of degree 3

About the group

Ring generators

Ring relations

Completion information

Restriction maps