I would like help in determining the proper notation to say:
The a group $G$ acting on a set of 3 points formed by the quotients $G/H$ where $H$ is a normal subgroup of $G$ is homomorphic to $S_3$
Thanks in advance!
I would like help in determining the proper notation to say:
The a group $G$ acting on a set of 3 points formed by the quotients $G/H$ where $H$ is a normal subgroup of $G$ is homomorphic to $S_3$
Thanks in advance!
Let G be a group isomorphic to $S_3$: $G \cong S_3$.
$H$ is a normal subgroup of G: $H \le G$
The group action: Define the map $f: G \rightarrow G/H$ by $g \mapsto gH$.
I would prefer to put it this way:
Let $G$ be a group. Let $H \subseteq G$ be a normal subgroup of $G$. (i.e.)Let $H \trianglelefteq G$ such that $|G/H|=3$. $G \circlearrowright G/H$
Why does this capture all the information?
Since, $H$ is a normal subgroup, $G/H$ is known to be a group and, we call it the quotient group. Ang given that $G$ acts on $G/H$ (i.e.) $G \circlearrowright G/H$, there is a canonical homomorphism, say $\varphi$ such that,
$\varphi:G \to \operatorname{Sym}(G/H) \cong S_3$
This conveys that $G$ is homomorphic to $S_3$.