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3 Automorphism groups and Canonical Forms

Sections

  1. Automorphism groups
  2. Canonical forms
  3. Example of canonical form computation

We refer to Eic07 for background on the algorithms used in this Chapter. Throughout the chapter, we assume that F is a finite field.

3.1 Automorphism groups

Let T be a nilpotent table over F. The following function can be used to determine the automorphism group of the algebra described by T. The automorphism group is determined as subgroup of GL(T.dim, T.fld) given by generators and its order. There is a variation available to determine the automorphism group of a modular group algebra FG, where F is a finite field and G is a p-group.

  • AutGroupOfTable( T ) F
  • AutGroupOfRad( FG ) F

    In both cases, the automorphism group is described by a record. The matrices in the lists glAutos and agAutos generate together the automorphism group. The matrices in agAutos generate a p-group. The entry size contains the order of the automorphism group.

    3.2 Canonical forms

    Let T be a nilpotent table. The following function can be used to determine the automorphism group of T if the underlying field of T is finite. The canonical form is a nilpotent table which is unique for the isomorphism type of the algebra defined by T. Again there a variation available for modular group algebras.

  • CanonicalFormOfTable( T ) F
  • CanonicalFormOfRad( FG ) F

    The automorphism group of T is determined as a side-product of computing the canonical form. The following functions can be used to return both.

  • CanoFormWithAutGroupOfTable( T ) F
  • CanoFormWithAutGroupOfRad( FG ) F

    In both cases, these functions return a record with entries cano and auto.

    3.3 Example of canonical form computation

    We compute the automorphism group and a canonical form for the modular group algebra of the dihedral group of order 8.

    gap> A := GroupRing(GF(2), SmallGroup(8,3));;
    gap> T := TableByWeightedBasisOfRad(A);;
    gap> C := CanoFormWithAutGroupOfTable(T);;
    
    # check that the canonical form is not equal to T
    gap> CompareTables(C.cano, T);
    false
    
    # the order of the automorphism group
    gap> C.auto.size;
    512
    
    # the entries of the canonical table as far as they are bounded
    gap> C.cano.tab;
    [ [ <a GF2 vector of length 7>, <a GF2 vector of length 7>, 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ], 
      [ <a GF2 vector of length 7>, <a GF2 vector of length 7>, 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ], 
      [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ], 
      [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] ], 
      [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] ], 
      [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ] ]
    

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    ModIsom manual
    January 2016