[ModIsom] 4 The modular isomorphism problem

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# 4 The modular isomorphism problem

### Sections

1. Computing and checking bins
2. Example for groups of order 64

An application of the methods in this package has been the checking of the modular isomorphism problems for the groups of order dividing 28, 36 and 29 Eic07,EKo10. This section contains the functions used for this purpose.

## 4.1 Computing and checking bins

- `BinsByGT( p, n ) F`

  returns a partion of the list [1·.NumberSmallGroups(pn)] into sublists so that the modular group algebras of two groups SmallGroup(pn, i) and SmallGroup(pn, j) can not be isomorphic if i and j are in different lists. The function BinsByGT uses various group theoretic invariants to split the groups of order pn in bins.

- `CheckBin( p, n, k, bin ) F`

  For i ∈ bin let Gi denote SmallGroup(pn, i) and let Ai be the augementation ideal of F Gi. This function computes and compares the canonical forms of the algebras Ai / Aij for every i ∈ bin and increasing j ∈ {1, …, k+1}.

  At each level j it splits the current bins into sub-bins according to the different canonical forms of Ai/Aij. Bins of length 1 are then discarded.

  The function returns if no further bins are available or if j=k+1 is reached. In the later case the function returns the remaining bins.

  ## 4.2 Example for groups of order 64

  We show how to check the modular isomorphism problem for the groups of order 64. We first use BinsByGT to determine bins and we then check the first of the resulting bins with CheckBin. The fact that CheckBin ends with an empty list of bins shows that all groups are splitted.

      gap> bins := BinsByGT(2,6);
      refine by abelian invariants of group (Sehgal/Ward)
      13 bins with 256 groups
      refine by abelian invariants of center (Sehgal/Ward)
      30 bins with 237 groups
      refine by lower central series (Sandling)
      32 bins with 127 groups
      refine by jennings series (Passi+Sehgal/Ritter+Sehgal)
      36 bins with 123 groups
      refine by conjugacy classes (Roggenkamp/Wursthorn)
      16 bins with 36 groups
      refine by elem-ab subgroups (Quillen)
        start bin 1 of 16
        start bin 2 of 16
        start bin 3 of 16
        start bin 4 of 16
        start bin 5 of 16
        start bin 6 of 16
        start bin 7 of 16
        start bin 8 of 16
        start bin 9 of 16
        start bin 10 of 16
        start bin 11 of 16
        start bin 12 of 16
        start bin 13 of 16
        start bin 14 of 16
        start bin 15 of 16
        start bin 16 of 16
      9 bins with 21 groups
      [ [ 13, 14 ], [ 18, 19 ], [ 20, 22 ], [ 97, 101 ], [ 108, 110 ],
        [ 155, 157, 159 ], [ 156, 158, 160 ], [ 173, 176 ], [ 179, 180, 181 ] ]

      gap> CheckBin(2,6,100,bins[1]);
      compute tables through power series
        determined table for 1
        determined table for 2

      refine bin
        weights yields bins [ [ 1, 2 ] ]
        layer 1 yields bins [ [ 1, 2 ] ]
        layer 2 yields bins [ [ 1, 2 ] ]
        layer 3 yields bins [ [ 1, 2 ] ]
        layer 4 yields bins [  ]

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