David J. Green

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Cohomology of group number 4 of order 125

About the group

Ring generators

Ring relations

Completion information

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General information on the group

• The group is also known as M125, the Extraspecial 5-group of order 125 and exponent 25.
• The group has 2 minimal generators and exponent 25.
• It is non-abelian.
• It has p-Rank 2.
• Its center has rank 1.
• It 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 1.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  1

  (t  −  1)2 · (t4  −  t3  +  t2  −  t  +  1) · (t4  +  t3  +  t2  +  t  +  1)

• The a-invariants are -∞,-2,-2. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 10:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. b_2_1, an element of degree 2
4. a_3_1, a nilpotent element of degree 3
5. a_5_1, a nilpotent element of degree 5
6. a_7_1, a nilpotent element of degree 7
7. a_9_1, a nilpotent element of degree 9
8. c_10_2, a Duflot regular element of degree 10

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Ring relations

There are 6 "obvious" relations:
   a_1_0², a_1_1², a_3_1², a_5_1², a_7_1², a_9_1²

Apart from that, there are 14 minimal relations of maximal degree 16:

1. b_2_1·a_1_0
2. a_1_0·a_3_1
3. b_2_1·a_3_1
4. a_1_0·a_5_1
5. b_2_1·a_5_1
6. a_3_1·a_5_1
7. a_1_0·a_7_1
8. b_2_1·a_7_1
9. a_3_1·a_7_1
10. a_1_0·a_9_1
11. a_5_1·a_7_1
12. a_3_1·a_9_1
13. a_5_1·a_9_1
14. a_7_1·a_9_1

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 16.
• 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_10_2, a Duflot regular element of degree 10
  2. b_2_1⁴, an element of degree 8
• The Raw Filter Degree Type of that HSOP is [-1, 8, 16].
• The filter degree type of any filter regular HSOP is [-1, -2, -2].
• We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 1 elements of degree 2.

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_2_1 → 0, an element of degree 2
4. a_3_1 → 0, an element of degree 3
5. a_5_1 → 0, an element of degree 5
6. a_7_1 → 0, an element of degree 7
7. a_9_1 → 0, an element of degree 9
8. c_10_2 →  − c_2_0⁵, an element of degree 10

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

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → a_1_1, an element of degree 1
3. b_2_1 → c_2_2, an element of degree 2
4. a_3_1 → 0, an element of degree 3
5. a_5_1 → 0, an element of degree 5
6. a_7_1 → 0, an element of degree 7
7. a_9_1 → c_2_2⁴·a_1_0 − c_2_1·c_2_2³·a_1_1, an element of degree 9
8. c_10_2 →  − c_2_2⁴·a_1_0·a_1_1 + c_2_1·c_2_2⁴ − c_2_1⁵, an element of degree 10

About the group

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