Quotients of permutation groups by nonabelian minimal normal subgroups

TL;DR

The paper proves that for a finite group $G\le \mathrm{Sym}(n)$ with a nonabelian minimal normal subgroup $N$, the quotient $G/N$ embeds into $\mathrm{Sym}(m)$ for some $m<n$, and when $G$ is transitive one can take $m\le \frac{2n}{5}$. It develops a toolkit of preliminary lemmas on normal subgroups, direct and subdirect products, and wreath products, and then conducts an induction on permutation degree, using the O'Nan-Scott classification to reduce to primitive almost simple or diagonal types. The main contribution is a uniform bound on the permutation degree of $G/N$ in terms of $n$, along with a sharper transitive bound, which yields corollaries about quotients with no abelian composition factors. These results have implications for algorithmic tasks such as finding small generating sets of finite groups and for understanding the structure of permutation group quotients in a concrete, degree-bounded way.

Abstract

We prove that a quotient G/N of a subgroup G of Sym(n) by a nonabelian minimal normal subgroup N of G embeds into Sym(m) for some $m<n$. This result was proved previously by Robert Chamberlain, and we also prove that,if G is transitive, then we can take m \le 2n/5.

Theorems & Definitions (18)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof