7.47 DimensionsLoewyFactors

DimensionsLoewyFactors( G )

Let G be p-group. Then DimensionsLoewyFactors returns the dimensions c_i of the Loewy factors of F_p<G>.

The Loewy series of F_p<G> is defined as follows. Let R be the Jacobson radical of the group ring F_p<G>. The series R^0 = F_p<G> > R^1 > ... > R^{l+1} = {1} is the Loewy series. The dimensions c_i are the dimensions of R^i / R^{i+1}.

    gap> f6 := FreeGroup( 6, "f6" );;
    gap> g := f6 / [ f6.1^3, f6.2^3, f6.3^3, f6.4^3, f6.5^3, f6.6^3,
    >          Comm(f6.3,f6.2)/f6.6^2,  Comm(f6.3,f6.1)/(f6.6*f6.5),
    >          Comm(f6.2,f6.1)/(f6.5*f6.4^2) ];;
    gap> a := AgGroupFpGroup(g);
    Group( f6.1, f6.2, f6.3, f6.4, f6.5, f6.6 )
    gap> DimensionsLoewyFactors(a);      
    [ 1, 3, 9, 16, 30, 42, 62, 72, 87, 85, 87, 72, 62, 42, 30, 16, 9, 3,
      1 ] 

The default function GroupOps.DimensionsLoewyFactors computes the Jennings series of G and uses Jennings thereom in order to calculate the dimensions of the Loewy factors.

Let G = X_1 geq X_2 geq ... geq X_l > X_{l+1}={1} be the Jennings series of G (see JenningsSeries) and let d_i be the dimensions of X_i / X_{i+1}. Then the Jennings polynomial is sum_i=0^l c_i x^i = prod_k=1^l (1+x^k+x^2k+...+x^(p-1)k)^d_k.

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