Normality of subgroups isn’t transitive

I’m trying to disprove the following statement: “let $$G$$ be a group, and suppose $$H,K$$ are subgroups of $$G$$; if $$H$$ is normal in $$G$$ and $$K$$ is normal in $$H$$, then $$K$$ is normal in $$G$$.”

My counterexample is with $$G=D_4$$, $$H=\{e,a^2,b,a^2b\}$$, and $$K=\{e,b\}$$.

I know that $$H$$ is normal in $$G$$ because $$[G:H]=2$$ and I know that $$K$$ is normal in $$H$$ because $$[H:K]=2$$. But we have that $$[G:K]=4$$, with $$b \notin Z(G)$$. So, $$K$$ is not normal in $$G$$. Therefore, normality of subgroups is not transitive. Is this correct?

• That's right, and this is one of the standard examples. Aug 13, 2020 at 6:07
• Another one is given in this post. There are several more posts about it, e.g., see here. Aug 13, 2020 at 10:53