Group Normalcy not transitive example?

[PsychonautQQ]

The normal subgroup of a normal subgroup need not be normal in the original group (normalcy is not transitive). Could somebody provide me with an example of where this is the case? Thanks :D

 

[mathwonk]

what examples have you tried? there should be some rather small suitable ones available.

 

[fresh_42]

Think of small symmetry groups or subgroups of them.

 

[PsychonautQQ]

Small symmetry groups... Is the Alternating symmetry group (A_n) always simple in S_n? If not that's the route I'm going to go... It's always normal because the index in S_n is always going to be two obviously... Anyway if I find a normal subgroup of some A_n maybe it won't be normal in S_n? Do you think this is a smart route to take?

 

[fresh_42]

Small symmetry groups... Is the Alternating symmetry group (A_n) always simple in S_n?

They are simple for $n > 4$ (and three). "Simple in" is a bit of a weird wording.

If not that's the route I'm going to go... It's always normal because the index in S_n is always going to be two obviously... Anyway if I find a normal subgroup of some A_n maybe it won't be normal in S_n? Do you think this is a smart route to take?

If you consider a whole symmetric group it might be more difficult to prove because you have an additional transposition at hand to get closure under conjugation.

 

[PsychonautQQ]

Ahh good point, my teacher had so engraved into me to make sure to say when a subgroup is normal it's very important to say what group it's normal inside of, but being simple is independent of that of course, good catch.

So don't consider the whole symmetric group? What do you mean i'll have an additional transposition? an additional transposition compared to the alternating group? I haven't spent a whole lot of time tinkering around with the inner workings of the symmetric group, are there certain elements in most or all symmetric groups that are often times an obvious normal subgroup that I could look for non-transitive normal subgroups of?

 

[fresh_42]

It's almost everything said already. Since the alternating groups $A_5, A_6, ...$ are all simple they won't help you. Abelian groups won't help you either. So to stay with small groups there is not much choice. My comment on the transposition just meant: With more possible $g$ in $gNg^{-1} ⊆ N$ it's not only more work to do, but also easier for a subgroup to be normal. However, I didn't really think a lot about it.

 

[Office_Shredder]

As you have observed constructing normal subgroups is not trivial, even in small groups to begin with. But if you have $$K\subset H \subset G$$ groups, with $$[G : H] = [H:K] = 2,$$ then K is a normal subgroup of H, and H is a normal subgroup of G, and you have a pretty good shot at K not being normal in G. K obviously needs to be non-trivial, so the smallest possible |G| is 8. It turns out that at least one of the two(?) non-abelian groups of order 8 works for this.