Confusing with the concept of normalizer $N_G(H)$ I'm Confusing with the concept of normalizer $N_G(H)$. It's a stupid question, sorry I'm new in this subject. Following the Hungerford's concept: > If $H$ acts by conjugation on the set $S$ of all subgroups of $G$, then the subgroup of $H$ fixing $K\in S$, namely $\\{h\in H\mid hKh^{-1}\\}$ is called the normalizer of $K$. Following the wikipedia's concept (regular concept) > The normalizer of S in the group G is defined to be > > $N_G(S)=\\{g\in G\mid gS=Sg \\} $ I'm a little confused, why in the Hungerford's concept, $H$ has to act by conjugation on the set $S$? these concepts are the equivalents? I need help here. Thanks in advance.

Well, saying $gS=Sg$ is the same as $gSg^{-1}=S$, so the definitions are equivalent.

The first definition simply emphasizes the fact that normalizers are just a special kind of stabilizers. The group $H$ in Hungerford's definition doesn't _have_ to act by conjugation as a prerequisite. Rather, we willingly _make_ it act by conjugation, and see what kind of stabilizers we will get.