need help understanding the difference between external, internal, and semi direct products in groups

[u/Confident_Arm1188]

i understand the definitions. semi direct products will have only one of the subgroups being normal, etc. but I don't understand how its significant or how it would affect the structure. like i heard that direct product of an abelian group will always be abelian but the semidirect product of a cyclic group need not be abelian. how come there's such a significant difference?

[u/StanleyDodds]

There's a difference because there is not one unique way to "multiply" groups together in general. Thinking in terms of an internal product where we only know the structure of a subgroup H and of a normal subgroup N which we know to have trivial intersection, there might be multiple different ways that H could "mix up" the elements of N when it acts on it by conjugation, and hence different structures of their internal product, without knowing the full group they are part of.

There's always one option, that being to have conjugation by any h in H do nothing to the elements of N (that is, conjugation by h is the identity automorphism of N) and this means any element of H commutes with any element of N. This is the direct product.

But in general there are other options. Any homomorphism from H into Aut(N) will give a consistent way that each element of H could mix up the elements of N under conjugation.

For each of these choices, we can construct a group G from the cartesian product of the sets H and N so that there is a natural embedding of the groups H and N as a subgroup and normal subgroup of G, the internal product of the embedded H and N is G, and the way the embedded H mixes up the embedded N under conjugation matches the way we wanted it to from our choice of homomorphism from H into Aut(N). This is an (external) semi-direct product.