I understand that someone has asked the definition of "an element of a group normalizes a subgroup". Let $G$ be a group and $a,b\in G$. Then what does "$a$ normalizes $b$" mean? Does it make sense at all, or does it just mean that $a$ commutes with $b$?
I came across this when I was reading the wikipedia article on Holomorph (mathematics).
There is one section as follows:
It says that "Such an $f$ normalizes any $\lambda(g)$, and the only $\lambda(g)$ that fixes the identity is $\lambda(1)$."
Here $g\in G$ and $\lambda:G\to \mathrm{Sym}(G)$; thus both $f$ and $\lambda(g)$ are elements (instead of subgroups) of $\mathrm{Sym}(G)$. Then what does "$f$ normalizes $\lambda(g)$" mean?