# Minimal finite groups with a given simple factor

Let $S$ be a non-abelian finite simple group. Call a finite group $G$ $S$-minimal if it admits $S$ as a Jordan-Hölder factor, but no proper subgroup of $G$ admits $S$ as a Jordan-Hölder factor.

For instance, $S$ itself is $S$-minimal, as well as any perfect central extension of $S$. Are these the only ones? Or what would be an example of $S$ and an $S$-minimal finite group $G$ that is not a central extension of $S$. If so, one can wonder whether in addition $d(G)<d(G')$ for every central extension $G'$ of $S$, where $d(H)$ denotes the smallest dimension of a faithful complex representation of $G$; this is asked here.

Remark: if $G$ is $S$-minimal, then $G$ has a unique maximal normal subgroup $N_G$ and $G/N_G$ is isomorphic to $S$. So I'm asking whether $N_G$ is always central in $G$.

Edit: the additional condition on $d$ cannot be achieved when $N_G$ is abelian.

Indeed, let $G$ be a finite group admitting $S$ as Jordan-Hölder factor, with $d(G)$ minimal; let us show that any normal abelian subgroup of $G$ is central.

Let $M$ be a normal subgroup of $G$ contained in $N_G$ (don't assume $M$ abelian for the moment). Let $V$ be a faithful $G$-module achieving $d(G)$. Clearly $V$ is irreducible, since otherwise we could find a group with $S$ as Jordan-Hölder factor with smaller $d(G)$. Now decompose $V$ according to distinct characters of $M$: $V=\bigoplus_{i\in I} V_i$, with $I\subset\hat{N}$. Then $G$ acts on $\hat{N}$, and preserves $I$; by irreducibility, the $G$-action on $I$ is transitive.

If $I$ is not a singleton, then the action of $G$ on $I$ factors through a nontrivial quotient of $G$, hence through a quotient $G'$ of $G$ admitting $S$ as Jordan-Hölder factor. Since $I$ has at most $n$ elements, this implies that $G'$ has a faithful module of dimension $\le n-1$, contradicting the minimality of $n$. Hence $I$ is a singleton.

If $N$ is abelian, this shows that $N$ acts on $V$ through a $G$-invariant character, and hence $N$ is central in $G$.

A nonsplit extension of an irreducible module by $S$ would also be $S$-minimal. For example, there is a non-split extension of $C_2^3$ by ${\rm SL}(3,2)$.
I don't know the answer to your question about whether $d(G) < d(G')$ for all central extensions $G'$, and I would very much like to. The smallest degrees of faithful representations of the finite quasisimple groups appear to be all known in all characteristics, and it would be very useful in a number of situations if it was true that there were no non-quasisimple $S$-minimal groups of this kind with smaller degree faithful representations.
• Thanks! it seems that this extension is just $SL_3(\mathbf{Z}/4\mathbf{Z})$.
• Note: this will never satisfy the addition condition about representations. Indeed, given a $n$-dim faithful rep $W$ of such $G$, let $N$ be the irreducible module, which is a normal abelian subgroup of $G$, and decompose $W$ as a direct sum $\bigoplus_{i\in I} W_i$ of eigenspaces of $N$. Then $G$ acts on $I$. If this action is nontrivial, this gives a nontrivial action of $S=G/V$ on $\le n$ elements and hence a faithful representation of $S$ in dimension $\le n-1$. If the $G$-action on $I$ is trivial, then $N$ is central in $G$.