Rank type conditions on commutators in finite groups

TL;DR

This work investigates how small-rank conditions on subgroups generated by commutators with Sylow subgroups constrain the overall rank of finite groups. By introducing I_G(S) and proving local–global rank bounds in p-soluble groups, it shows that, under a local r-generated condition on I_G(P), the subgroup [G,P] has r-bounded rank; with stronger local hypotheses, this extends to arbitrary finite groups and to G' under a global condition across all primes. A key tool is a rank-bound result for coprime automorphism groups A acting on G via I_G(A). The paper also furnishes counterexamples when p-solubility is absent, and it provides detailed proofs across nilpotent, semisimple, and soluble cases, relying on the Classification of finite simple groups and representation-theoretic methods. The results contribute a robust local-to-global framework controlling ranks of commutator-related subgroups in finite groups with and without solubility assumptions.

Abstract

For a subgroup $S$ of a group $G$, let $I_G(S)$ denote the set of commutators $[g,s]=g^{-1}g^s$, where $g\in G$ and $s\in S$, so that $[G,S]$ is the subgroup generated by $I_G(S)$. We prove that if $G$ is a $p$-soluble finite group with a Sylow $p$-subgroup $P$ such that any subgroup generated by a subset of $I_G(P)$ is $r$-generated, then $[G,P]$ has $r$-bounded rank. We produce examples showing that such a result does not hold without the assumption of $p$-solubility. Instead, we prove that if a finite group $G$ has a Sylow $p$-subgroup $P$ such that (a) any subgroup generated by a subset of $I_G(P)$ is $r$-generated, and (b) for any $x\in I_G(P)$, any subgroup generated by a subset of $I_G(x)$ is $r$-generated, then $[G,P]$ has $r$-bounded rank. We also prove that if $G$ is a finite group such that for every prime $p$ dividing $|G|$ for any Sylow $p$-subgroup $P$, any subgroup generated by a subset of $I_G(P)$ can be generated by $r$ elements, then the derived subgroup $G'$ has $r$-bounded rank. As an important tool in the proofs, we prove the following result, which is also of independent interest: if a finite group $G$ admits a group of coprime automorphisms $A$ such that any subgroup generated by a subset of $I_G(A)$ is $r$-generated, then the rank of $[G,A]$ is $r$-bounded.

Theorems & Definitions (85)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5