A finite group that splits and does not split

Question

Is there an example of a finite group $A$ that acts on a finite group $C$ irreducibly (that is, $C$ has no proper nontrivial $A$-invariant subgroup) such that there exists an epimorphism $$\tau \colon A \ltimes C \to A$$ which does not split?

Answer 1

I don't believe this is possible. Let $G$ be a minimal counterexample. Then $G = AC$ with $C \unlhd G$ and $A$ a complement of $C$, and there exists $D \unlhd G$ with $G/D \cong A$, but $D$ has no complement if $G$. Note that $G/D \cong G/C \Rightarrow |C|=|D|$. Since $A$ acts irreducibly on $C$, we have $C \cap D = 1$.

We prove first that $D \le A$. Since $C \cap D = 1$, we have $CD \cap A \cong D$, and $CD \cap A$ is a subgroup of $C \times D$ of the form $\{(\phi(d),d) : d \in D \}$, where $\phi:D \to C$ is a homomorphism.

But $CD \cap A \unlhd A$, and for $a \in A$ (with notation $g^h=h^{-1}gh$) we have $(\phi(d),d)^a = (\phi(d)^a, d^a)$, but this must also equal $(\phi(d^a),d^a)$, so $\phi(d^a) = \phi(d)^a$ and hence ${\rm Im}(\phi)$ is $A$-invariant so, by assumption, it must be $C$ or $1$.

If ${\rm Im}(\phi) = C$ then $\ker(\phi)=1$, so $D \cap A = 1$. But then $A$ is a complement of $D$ in $G$, contrary to assumption. So $ {\rm Im}(\phi) = 1$ and $D \le A$ as claimed.

So now $G/D$ is a split extension of $C$ by $A/D$ and, since $D$ centralizes $C$, $A/D$ must act irreducibly on $C$.

If $D$ had a complement $B$ in $A$, then $CB$ would be a complement of $D$ in $G$, contrary to assumption. So $G/C \cong A$ is a nonsplit extension of $D$ by $A/D$. But we are assuming that $G/C \cong G/D$, so $G/C$ is a smaller counterexample than $G$.

Answer 2

I take it Pablo your question can be rephrased as follows. Does there exist an epimorphism $\tau\colon A\ltimes C\to A$ where $A$ acts irreducibly on $C$ and where $\ker(\tau)\ne C$? If this is your question and $A$ can act non-faithfully, then the answer is Yes. Take As $\tau\colon{\rm C}_6\times {\rm C}_3\to {\rm C}_6$ where $\ker(\tau)={\rm C}_3\leq{\rm C}_6$. I suspect you want $A$ to act faithfully, see below.

As $K:=\ker(\tau)$ and $C$ are $A$-invariant, so is $K\cap C$. Assume $K\ne C$. Then $K\cap C=1$ by $A$-irreducibility. It follows from $C\trianglelefteq A\ltimes C$ that $\tau(C)\trianglelefteq\tau(A\ltimes C)$ or $C\trianglelefteq A$. However, $C$ is characteristically simple, by $A$-irreducibility, so it equals $T^n$ where $T$ is a finite simple group. The split extension is associated with a (nontrivial) homomorphism $\phi\colon A\to{\rm Aut}(C)$, and ${\rm Aut}(C)={\rm Aut}(T^n)$ equals ${\rm GL}_n(p)$ or ${\rm Aut}(T)\wr{\rm Sym}_n$ depending on whether or not $T$ is abelian. If $C={\rm C}_2^4$ and $A={\rm C_5}\ltimes C$ with ${\rm C}_5$ acting irreducibly, then the epimorphism $\tau\colon A\ltimes C\to A$ with $\ker(\tau)=C\leq A$ is such a "nonsplit" example. If $\phi$ is faithful, then $C=T^n\trianglelefteq A\leq T^n\rtimes{\rm Sym}_n$ or ${\rm GL}_n(p)$. Is this the problem you wanted to explore?