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5 Covering Subgroups

Let X be a collection of groups closed under taking homomorphic images. An X-covering subgroup of a group G is a subgroup E satisfying

(C) E is in X , and EV = U whenever E is contained in U is contained in G with U/V in X.

It follows from the definition that an X-covering subgroup E of G is also X-covering in every subgroup U of G that contains E, and an easy argument shows that E is an X-projector of every such U, i.e., E satisfies

(P)   EK/K is an X-maximal subgroup of U/K whenever K is normal in U.

Gaschütz showed that if F is a locally defined formation, then every finite solvable group has an F-covering subgroup. Indeed, locally defined formations are the only formations with this property. For such formations the F-projectors and F-covering subgroups of a solvable group coincide and form a single conjugacy class of subgroups. (See DH for details.)

  • CoveringSubgroup1( G, F ) O
  • CoveringSubgroup2( G, F ) O
  • CoveringSubgroupWrtFormation( G, F ) O

    If F is a locally defined integrated formation in GAP and if G is a finite solvable group, then the command CoveringSubgroup1( G, F ) returns an F-covering subgroup of G. The function CoveringSubgroup2 uses a different algorithm to compute F-covering subgroups. The user may choose either function. Experiments with large groups suggest that CoveringSubgroup1 is somewhat faster. CoveringSubgroupWrtFormation checks first to see if either of these two functions has already computed an F-covering subgroup of G and, if not, it calls FCoveringGroup1 to compute one.

    Nilpotent-covering subgroups are also called Carter subgroups.

  • CarterSubgroup( G ) A

    The command CarterSubgroup( G ) is equivalent to CoveringSubgroupWrtFormation( G, Formation( "Nilpotent" ) ).

    All of these functions call upon F-normalizer algorithms as subroutines.

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    November 2011