# Is $W_G(S) = \{ g\in G \mid \exists s\in S, \ gsg^{-1}\in S \}$ well-known and studied?

Given a group $$G$$ and a subset of the group $$S$$, the normalizer $$N_G(S)$$ is defined as the set $$N_G(S) = \{ g\in G \mid \forall s\in S, \ gsg^{-1}\in S \}$$

Is a weaker version of that (which is certainly not a group), such that $$W_G(S) = \{ g\in G \mid \exists s\in S, \ gsg^{-1}\in S \}$$ well-known and studied?

• What do you mean by "is there... ?", do you mean it in a "was that ever studied?" way perhaps? Commented May 8, 2023 at 20:57
• Yes. Is this concept well understood, does it have nice properties, interesting applications, etc. Basically I just want an answer of the type, "yes, this is called the ____ , see source ____". Commented May 8, 2023 at 21:23
• If $e\in S$, then $W_G(S)=G$. Commented May 8, 2023 at 21:54
• Extending the comment of @Shaun : if $S \cap Z(G) \neq \emptyset$, then $W_G(S)=G$. Commented May 8, 2023 at 22:18
• To generalize the observation even more, if $S$ contains a conjugacy class, then $W_G(S)=G$. this includes the case of $S$ containing a central element, since central elements are their own conjugacy class. Commented May 9, 2023 at 1:03