[GAP Forum] Computing a nonabelian tensor square

[Bettina Eick]

Dear Arturo,

> Let Z be the infinite cyclic group, and let G be the semidirect product of Z 
> with itself, with Z acting nontrivially; that is,
>
> G = < x,y  |   x^y = x^{-1} >
>
> I would like to check whether the nonabelian tensor square of G has torsion, 
> and to confirm that the nonabelian tensor square of G/Z(G) does in fact have 
> torsion. (G/Z(G) is the infinite dihedral group, as Z(G) = <y^2> ).

This can be done as follows:

# create G as group with power-conjugate presentation
gap> c := FromTheLeftCollector(2);
<<from the left collector with 2 generators>>
gap> SetConjugate(c, 2, 1, [2,-1]);
gap> UpdatePolycyclicCollector(c);
gap> G := PcpGroupByCollector(c);
Pcp-group with orders [ 0, 0 ]

# compute the non-abelian tensor square 
gap> H := NonAbelianTensorSquare(G);
Pcp-group with orders [ 0, 0, 2, 4 ]
gap> TH := TorsionSubgroup(H);
Pcp-group with orders [ 2, 4 ]
gap> Size(TH);
8

# one can also investigate H further
gap> IsAbelian(H);
true
gap> AbelianInvariants(H);
[ 0, 0, 2, 4 ]

# and one can do a similar computation with G/Center(G)
gap> K := G/Center(G);
Pcp-group with orders [ 2, 0 ]
gap> L := NonAbelianTensorSquare(K);
Pcp-group with orders [ 0, 2, 2, 2 ]
gap> IsAbelian(L);
true
gap> AbelianInvariants(L);
[ 0, 2, 2, 2 ]

Best wishes,

Bettina