Question:Normal subgroup of normal subgroup big deal

Q: '''I was told that it is very important and counter-intuitive that a normal subgroup of a normal subgroup need not be normal, but I didn't find it either important or counter-intuitive. Why is it true and What is the significance?'''

A: You're referring to the fact that answer references::normality is not transitive, and hence that a answer references::2-subnormal subgroup (and more generally a answer references::subnormal subgroup) need not be normal. This is indeed important, though it is not necessarily counter-intuitive. The importance is partly because if the opposite were true, it would prove a very convenient way of showing that subgroups are normal, and thus make group theory very different and perhaps more boring.

One way of thinking about the significance is to look at the implications for quotient groups. If $$H \le K \le G$$ with $$H$$ normal in $$K$$ and $$K$$ normal in $$G$$, we can talk of the quotient groups $$G/K$$ and $$K/H$$. If it were also true that $$H$$ is normal in $$G$$, we'd have a group $$G/H$$, whereby $$K/H$$ could be identified with a normal subgroup of it and the quotient would be isomorphic to $$G/K$$ (by the answer references::third isomorphism theorem). However, the point is that we are not guaranteed that $$H$$ is normal in $$G$$. This fact shows that there is, in some sense, a lot more flexibility in the way groups can be put together.