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 $g \mapsto gH$.

  • 3
    $\begingroup$ $\leq$ is standard notation for "is a subgroup of", not "is a normal subgroup of", which is usually denoted $\trianglelefteq$ $\endgroup$ – Tobias Kildetoft Jan 28 '12 at 10:07

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 $S_3$.

  • $\begingroup$ I don't like this (although it can be saved!) for the simple reason you aren't using proper sentences. Are you saying if $H\unlhd G$ then $|G/H|=3$? Obviously not...but only 'cause I know you don't mean that! You need to be more clear. $\endgroup$ – user1729 Jan 28 '12 at 21:05
  • $\begingroup$ (Also, it is perhaps useful to point out that there is no standard notation for a group action. See this mathoverflow discussion, mathoverflow.net/questions/64544/…) $\endgroup$ – user1729 Jan 28 '12 at 21:06
  • $\begingroup$ @user1729 I have never used a then. But, I'll be more explicit right away! And, for the notation on action, the OP never asked for a notation to use in papers or sth. I assume that he will use it for his notes. $\endgroup$ – user21436 Jan 28 '12 at 21:07

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