73.76 AllCat1s

AllCat1s( G )

By a em cat1-structure on G we mean a cat1-group {cal C} where R is a subgroup of G and e is the inclusion map. For such a structure to exist, G must contain a normal subgroup S with G/S cong R. Furthermore, since t,h are respectively the identity and zero maps on S, we require R cap S = { 1_G }. This function uses EndomorphismClasses( G, 3 ) (see refEndomorphismClasses, refIdempotentImages) to construct idempotent endomorphisms of G as potential tails and heads. A backtrack procedure then tests to see which pairs of idempotents give cat1-groups. A non-documented function AreIsomorphicCat1s is called in order that the function returns representatives for isomorphism classes of cat1-structures on G. See xmodAW1 for all cat1-structures on groups of order up to 30.

    gap> AllCat1s( a4 );
    There are 1 endomorphism classes.
    Calculating idempotent endomorphisms.
    # idempotents mapping to lattice class representatives:
    [ 1, 0, 1, 0, 1 ]
    Isomorphism class 1
    : kernel of tail = [ "2x2" ]
    :    range group = [ "3" ]
    Isomorphism class 2
    : kernel of tail = [ "1" ]
    :    range group = [ "A4" ]
    [ cat1-group [a4 ==> a4.H3] , cat1-group [a4 ==> a4] ]  

The first class has range c3 and kernel k4. The second class contails all cat1-groups {cal C} = (alpha^{-1}; alpha, alpha : G to G) where alpha is an automorphism of G.

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GAP 3.4.4
April 1997