An application of the methods in this package has been the checking
of the modular isomorphism problems for the groups of order dividing
2^{8}, 3^{6} and 2^{9} Eic07,EKo10. This section contains the
functions used for this purpose.

`BinsByGT( p, n ) F`

returns a partion of the list [1·.*NumberSmallGroups*(*p*^{n})] into
sublists so that the modular group algebras of two groups
SmallGroup(*p*^{n}, *i*) and SmallGroup(*p*^{n}, *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 *p*^{n} in bins.

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

For *i* ∈ *bin* let *G*_{i} denote SmallGroup(*p*^{n}, *i*) and let *A*_{i}
be the augementation ideal of *F* *G*_{i}. This function computes and
compares the canonical forms of the algebras *A*_{i} / *A*_{i}^{j} 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 *A*_{i}/*A*_{i}^{j}. 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.

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 [ ]

[Up] [Previous] [Next] [Index]

ModIsom manual

January 2020