Hi,

Can the same semidirect product represent different non-isomorphic groups?

For this seems to be the case in a group of order 130 for instance. This groups has 4 non-isomorphic cases: one cyclic and the others non abelian.

In these 3 others, the subgroup of order 65 (say H65) is cyclic and normal while the subgroup of order 2 (say H2) is cyclic and not normal.

Furthermore, their intersection is trivial.

And noting that H65H2 is a subgroup of G and that |H65H2|=|H65||H2|/1=|G|, then G = H65H2, hence G is the semidirect product of H65 and H2.

Hence, the semidirect product of Z65xZ2 seems to represent 3 different non-isomrphic cases. Is that correct?

Another thing: the dihedral group D65 can be viewed as a semidirect product of Z65xZ2. But is D65 isomorphic to Z65xZ2? Because it is definitely wrong that D65 represents 3 non-somorphic cases.

So, what I beleive is that the semidirect product Z65xZ2 indeed represents 3 different non-isomorphic cases, one of which is D65.

Am I correct?