[Math] Does the operation of $G/A$ on $A$ by conjugation just give a trivial homomorphism


I'm having doubts about an exercise, since I reach a trivial conclusion. I hope it's not an issue to see if I'm screwing up.

Let $G$ be a group, and $A$ a normal abelian subgroup. Show $G/A$ operates on $A$ by conjugation and in this way get a homomorphism of $G/A$ into $\text{Aut}\ (A)$.

So the action is $gA\bullet a=gAa(gA)^{-1}=gAag^{-1}A=gAaAg^{-1}=gAg^{-1}=A$. Also $(gAhA)\bullet a=gA\bullet(hA\bullet a)$ and $A\bullet a=A$. So is this just a trivial homomorphism which maps everything to $A$?

I think I'm applying things wrong since I thought it's supposed to be $A\bullet a=a$ and the action should return elements of $A$, not $G/A$. How to interpret correctly?

Best Answer

Remember that any action of a group $G$ on a set $X$ induces a homomorphism $G\to S_X$, the group of permutations on $X$; and conversely, that any group homomorphism $G\to S_X$ induces an action of $G$ on $X$. Given an action $G\times X\to X$, the homomorphism $\varphi\colon G\to S_X$ maps $g$ to $\varphi(g)$, the function such that $\varphi(g)(x) = g\cdot x$. Conversely, given a homorphism $\psi\colon G\to S_X$, $\psi$ defines an action by $g\cdot x = \psi(g)(a)$.

With that preliminary, let let $G$ act on $A$ by conjugation in the usual way: $g\cdot a= gag^{-1}\in A$ (it's in $A$ because $A$ is normal). This is an action of $G$ on $A$.

This induces a homomorphism $G\to S_A$, where we consider $A$ as a set.

However, because $A$ is abelian, for every $a\in A$ and every $b\in A$ we have $b\cdot a = a$. That is, the action of $A$ on itself is the identity permutation, so the homomorphism $G\to S_A$ factors through $G/A$. This gives a homomorphism $G/A \to S_A$. And of course, any homomorphism from a group $H$ to a permutation group $S_X$ induces an action of $H$ on $X$, so $G/A$ acts on $A$.

The action of $G/A$ on $A$ is as follows: given $gA\in G/A$ and $a\in A$, $gA\cdot a = gag^{-1}$. This is well defined, because if $gA=hA$, then $h^{-1}g\in A$, so $(h^{-1}g)a(g^{-1}h) = a$, hence $gag^{-1}=hah^{-1}$.

The action is not trivial unless $A$ is central.

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