7.22 DerivedSubgroup

DerivedSubgroup( G )

DerivedSubgroup returns the derived subgroup <G>^prime = [ <G>, <G> ] of G.

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

Note that DerivedSubgroup sets and tests G.derivedSubgroup. CommutatorSubgroup (see CommutatorSubgroup) allows you to compute the commutator group of two subgroups.

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

Let G be generated by g_1, ..., g_n. Then the default function GroupOps.DerivedSubgroup returns the normal closure of S under G where S is the subgroup of G generated by Comm( g_i, g_j ) for 1 leq j < i leq n.

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