SubXMod( X, subS, subR )
A sub-crossed module of a crossed module X
has as source a subgroup
subS
of X.source
and as range a subgroup subR
of X.range
. The
boundary map and the action are the appropriate restrictions. In the
following example we construct a sub-crossed module of SX
with range
q8
and source a cyclic group of order 4.
gap> q8 := SX.source;; genq8 := q8.generators;; gap> q8.name := "q8";; XModName( SX );; gap> c4 := Subgroup( q8, [ genq8[1] ] ); Subgroup( sl(2,3), [ (1,2,3,4)(5,8,7,6) ] ) gap> c4.name := "c4";; gap> subSX := SubXMod( SX, c4, q8 ); Crossed module [c4->q8] gap> XModPrint( subSX ); Crossed module [c4->q8] :- : Source group has parent ( sl(2,3) ) and has generators: [ (1,2,3,4)(5,8,7,6) ] : Range group has parent ( sl(2,3) ) and has generators: [ (1,2,3,4)(5,8,7,6), ( 1, 5, 3, 7)( 2, 6, 4, 8) ] : Boundary homomorphism maps source generators to: [ ( 1, 2, 3, 4)( 5, 8, 7, 6) ] : Action homomorphism maps range generators to automorphisms: (1,2,3,4)(5,8,7,6) --> {source gens --> [ (1,2,3,4)(5,8,7,6) ]} (1,5,3,7)(2,6,4,8) --> {source gens --> [ (1,4,3,2)(5,6,7,8) ]} These 2 automorphisms generate the group of automorphisms.
GAP 3.4.4