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.
GAP 3.4.4