Cohomology of group number 4 of order 343

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343

General information on the group

  • The group is also known as M343, the Extraspecial 7-group of order 343 and exponent 49.
  • The group has 2 minimal generators and exponent 49.
  • 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 · (t6  −  t5  +  t4  −  t3  +  t2  −  t  +  1) · (t6  +  t5  +  t4  +  t3  +  t2  +  t  +  1)
  • The a-invariants are -∞,-2,-2. They were obtained using the filter regular HSOP of the Benson test.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343

Ring generators

The cohomology ring has 10 minimal generators of maximal degree 14:

  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. a_11_1, a nilpotent element of degree 11
  9. a_13_1, a nilpotent element of degree 13
  10. c_14_2, a Duflot regular element of degree 14

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343

Ring relations

There are 8 "obvious" relations:
   a_1_02, a_1_12, a_3_12, a_5_12, a_7_12, a_9_12, a_11_12, a_13_12

Apart from that, there are 27 minimal relations of maximal degree 24:

  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. b_2_1·a_9_1
  12. a_5_1·a_7_1
  13. a_3_1·a_9_1
  14. a_1_0·a_11_1
  15. b_2_1·a_11_1
  16. a_5_1·a_9_1
  17. a_3_1·a_11_1
  18. a_1_0·a_13_1
  19. a_7_1·a_9_1
  20. a_5_1·a_11_1
  21. a_3_1·a_13_1
  22. a_7_1·a_11_1
  23. a_5_1·a_13_1
  24. a_9_1·a_11_1
  25. a_7_1·a_13_1
  26. a_9_1·a_13_1
  27. a_11_1·a_13_1

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 24.
  • 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_14_2, a Duflot regular element of degree 14
    2. b_2_16, an element of degree 12
  • The Raw Filter Degree Type of that HSOP is [-1, 12, 24].
  • 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.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343

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. a_11_1 → 0, an element of degree 11
  9. a_13_1 → 0, an element of degree 13
  10. c_14_2 →  − c_2_07, an element of degree 14

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 → 0, an element of degree 9
  8. a_11_1 → 0, an element of degree 11
  9. a_13_1 → c_2_26·a_1_0 − c_2_1·c_2_25·a_1_1, an element of degree 13
  10. c_14_2 →  − 2·c_2_26·a_1_0·a_1_1 + c_2_1·c_2_26 − c_2_17, an element of degree 14

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343


Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 25.08.2009