24.1 More about Ag Words

Let G be a group and G = G_0 > G_1 > ... > G_n = 1 be a subnormal series of G neq 1 with finite cyclic factors, i.e., G_i lhd G_{i-1} for all i=1, ..., n and G_{i-1} = < G_i, g_i > . Then G will be called an ag group indexag group with AG generating sequence indexAG generating sequence or, for short, AG system / G_i. If all o_1, ..., o_n are primes the system (g_1, ..., g_n) is called a *PAG system* index{PAG system}. With respect to a given AG system the group G has a so called *power-commutator presentation*

begintabularlcll {g_i}^{o_i} & = & w_{ii}(g_{i+1},..., g_n) & for 1leq ileq n,
[g_i,g_j] & = & w_{ij}(g_{j+1},...,g_n) & for 1leq j< ileq n
endtabular

and a so called power-conjugate presentation

begintabularlcll {g_i}^{o_i} & = & w_{ii}(g_{i+1},..., g_n) & for 1leq ileq n,
g_i^{g_j} & = & w^{prime}_{ij}(g_{j+1},...,g_n) & for 1leq j< ileq n.
endtabular

For both kinds of presentations we shall use the term AG presentation. Each element g of G can be expressed uniquely in the form

begintabularcc g = g_1^{nu_1}* ...* g_n^{nu_n} & for 0 leq nu_i < o_i. endtabular

We call the composition series G_0 > G_1 > ... > G_n the AG series of G and define nu_i( g ) := nu_i. If nu_i = 0 for i = 1, ..., k-1 and nu_k neq 0, we call nu_k the leading exponent and k the depth of g and denote them by nu_k =: lambda( g ) and k =: delta( g ). We call o_k the relative order of g.

Each element g of G is called ag word and we say that G is the parent group of g. A parent group is constructed in GAP using AgGroup (see AgGroup) or AgGroupFpGroup (see AgGroupFpGroup).

Our standard example in the following sections is the symmetric group of degree 4, defined by the following sequence of GAP statements. You should enter them before running any example. For details on AbstractGenerators see AbstractGenerator.

    gap> a  := AbstractGenerator( "a" );;  # (1,2)
    gap> b  := AbstractGenerator( "b" );;  # (1,2,3)
    gap> c  := AbstractGenerator( "c" );;  # (1,3)(2,4)
    gap> d  := AbstractGenerator( "d" );;  # (1,2)(3,4)
    gap> s4 := AgGroupFpGroup( rec(
    >        generators := [ a, b, c, d ],
    >        relators   := [ a^2, b^3, c^2, d^2, Comm( b, a ) / b,
    >                        Comm( c, a ) / d, Comm( d, a ),
    >                        Comm( c, b ) / ( c*d ), Comm( d, b ) / c,
    >                        Comm( d, c ) ] ) );
    Group( a, b, c, d )
    gap> s4.name := "s4";;
    gap> a := s4.generators[1];; b := s4.generators[2];;
    gap> c := s4.generators[3];; d := s4.generators[4];; 

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