$H \subseteq G$. About the embedding(s?) of $\operatorname{Sym}(H)$ into $\operatorname{Sym}(G)$

abstract-algebragroup-homomorphismgroup-theory

Let $G$ be a set and $H \subseteq G$. Given a bijection $u$ on $G \setminus H$, any bijection $\alpha$ on $H$ can be extended to a bijection $\varphi_u(\alpha)$ on $G$ by:

\begin{alignat}{1}
&\varphi_u(\alpha)_{|H}:=\alpha \\
&\varphi_u(\alpha)_{|G \setminus H}:=u \\
\end{alignat}

I note that $\varphi_u(\alpha)=\varphi_u(\beta) \Rightarrow \varphi_u(\alpha)_{|H}=\varphi_u(\beta)_{|H} \Rightarrow \alpha=\beta$, so that $\varphi_u$ is injective for all $u \in \operatorname{Sym}(G \setminus H)$. Differently,

$$\varphi_u(\alpha\beta)=\varphi_u(\alpha)\varphi_u(\beta) \Leftrightarrow u=\iota_{G \setminus H} \tag 1$$

If so, we'd have that among all $\varphi_u$'s, just $\varphi_{\iota_{G \setminus H}}$ is an embedding of $\operatorname{Sym}(H)$ into $\operatorname{Sym}(G)$. I use the conditional, because I'm not sure about $(1)$: is it correct? If so, how can I find (all the) other embeddings of $\operatorname{Sym}(H)$ into $\operatorname{Sym}(G)$?

Edit

Remark 1 – Embeddings of $\operatorname{Sym}(H)$ into $\operatorname{Sym}(G)$ are equivalent to faithful actions of $\operatorname{Sym}(H)$ on $G$.

Remark 2 – If we define:

$$\operatorname{Sym}_G(H):=\{f \in \operatorname{Sym}(G) \mid f_{|H} \in \operatorname{Sym}(H)\}$$

we have that:

  • $\operatorname{Sym}_G(H) \le \operatorname{Sym}(G)$;
  • the map $\psi \colon \operatorname{Sym}_G(H) \rightarrow \operatorname{Sym}(H)$, $f \mapsto \psi_f:=f_{|H}$ is epimorphism; then, for the First Homomorphism Theorem, we get:

$$\operatorname{Sym}(H) \cong \operatorname{Sym}_G(H)/\operatorname{ker}(\psi)$$

where $\operatorname{ker}(\psi)=\{f \in \operatorname{Sym}_G(H) \mid f_{|H}=\iota_{\operatorname{Sym}(H)}\}$.

Best Answer

Yes. In other words, given $x \in G\setminus H$, if we have $u(x) = \varphi_u(\alpha\beta)(x) = \varphi_u(\alpha)\circ \varphi_u(\beta)(x) = u(u(x))$, we can conclude $u(x) = x$ for all $x \in G\setminus H$. You should argue why this is true.

As far as finding other embeddings of Sym($H$) in Sym($G$), what you have shown is that in this construction the embedding depends solely on the embedding of $H$ into $G$. Can you use this to find all of them arising this way?

Edit: as Derek Holt points out in the comments, not all embeddings Sym($H$) into Sym(G) arise in this way. For instance, if $|G| = 2|H|$, then you can have Sym($H$) act on two disjoint subsets of $G$ at once.

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