25.86 OneCoboundaries

OneCoboundaries( G, M )

Let M be a normal p-elementary abelian subgroup of G. Then OneCoboundaries computes the vector space {cal V} = beta( B^1( <G>/<M>, <M> ) ), which is isomorphic to the group of one coboundaries B^1( G, M ) as described in One Cohomology Group. The functions returns a record C with the following components.

oneCoboundaries:

contains the vector space {cal V}.

generators:

contains representatives (g_1, ..., g_l) for the canonical generating system of <G> / <M>

cocycleToList:

contains a functions which takes an element v of {cal V} as argument and returns a list [ n_1, ..., n_l ], where n_i is an element of M, such that n_i = ( beta^{-1}( v ) )( g_i <M> ).

listToCocycles:

is the inverse of cocycleToList.

OneCoboundaries( G, alpha, M )

In that form OneCoboundaries computes the one coboundaries in the semidirect product of G and M where G acts on M using alpha (see SemidirectProduct).

    gap> s4xc2 := DirectProduct( s4, CyclicGroup( AgWords, 2 ) );
    Group( a1, a2, a3, a4, b )
    gap> m := CompositionSubgroup( s4xc2, 3 );
    Subgroup( Group( a1, a2, a3, a4, b ), [ a3, a4, b ] )
    gap> oc := OneCoboundaries( s4xc2, m );
    rec(
      oneCoboundaries := RowSpace( GF(2),
        [ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
          [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ] ] ),
      generators := [ a1, a2 ],
      cocycleToList := function ( c ) ... end,
      listToCocycle := function ( L ) ... end )
    gap> v := Base( oc.oneCoboundaries );
    [ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
      [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ] ]
    gap> oc.cocycleToList( v[1] );
    [ a4, a4 ]
    gap> oc.cocycleToList( v[2] );
    [ IdAgWord, a3 ]
    gap> oc.cocycleToList( v[1]+v[2] );
    [ a4, a3*a4 ] 

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