7.53 IsCentral

IsCentral( G, U )

IsCentral returns true if the group G centralizes the group U and false otherwise.

A group G centralizes a group U if and only if for all g in <G> and for all u in <U> the equation g* u = u* g holds. Note that U need not to be a subgroup of G but they must have a common parent group.

Note that IsCentral sets and tests U.isCentral if G is the parent group of U.

    gap> s4 := Group( (1,2,3,4), (1,2) );;
    gap> d8 := Subgroup( s4, [ (1,2,3,4), (1,2)(3,4) ] );;
    gap> c2 := Subgroup( s4, [ (1,3)(2,4) ] );;
    gap> IsCentral( s4, c2 );
    false
    gap> IsCentral( d8, c2 );
    true 

The default function GroupOps.IsCentral tests whether G centralizes U by testing whether the generators of G commutes with the generators of U.

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