7.19 CommutatorSubgroup

CommutatorSubgroup( G, H )

Let G and H be groups with a common parent group. CommutatorSubgroup returns the commutator subgroup [ G, H ].

The commutator subgroup of G and H is the group generated by all commutators [ g, h ] with g in <G> and h in <H>.

See also DerivedSubgroup (DerivedSubgroup).

    gap> s4 := Group( (1,2,3,4), (1,2) );
    Group( (1,2,3,4), (1,2) )
    gap> d8 := Group( (1,2,3,4), (1,2)(3,4) );
    Group( (1,2,3,4), (1,2)(3,4) )
    gap> CommutatorSubgroup( s4, AsSubgroup( s4, d8 ) );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (1,3)(2,4), (1,3,2) ] ) 

Let G be generated by g_1, ..., g_n and H be generated by h_1, ..., h_m. The normal closure of the subgroup S generated by Comm( g_i, h_j ) for 1 leq i leq n and 1 leq j leq m under G and H is the commutator subgroup of G and H (see Hup67). The default function GroupOps.CommutatorSubgroup returns the normal closure of S under the closure of G and H.

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