[Paper Review] Powers in finite groups

Dan Segal, Nikolay Nikolov|ArXiv.org|Sep 25, 2009

Finite Group Theory Research4 references6 citations

TL;DR

This paper establishes that in any finite group with d generators, every element of the verbal subgroup G^q (generated by qth powers) can be expressed as a product of at most f(d,q) qth powers, where f(d,q) depends only on d and q. This implies that G^q is open in finitely generated profinite groups, extending the solution to the Restricted Burnside Problem and confirming that qth power verbal subgroups have finite width and are closed in such groups.

ABSTRACT

In this note we prove that if $G$ is a finitely generated profinite group then the verbal subgroup $G^{q}$ is open. Equivalently in a $d$-generator finite group every product of $q$th powers is a product of $f(d,q)$ $q$th powers.

Motivation & Objective

• To establish a uniform bound f(d,q) on the number of qth powers needed to generate any element of G^q in a d-generator finite group.
• To prove that the verbal subgroup G^q is open in any finitely generated profinite group, extending the solution to the Restricted Burnside Problem.
• To generalize earlier results on verbal subgroups being closed in profinite groups, particularly for power words x^q.
• To show that the word x^q is d-restricted for every d, implying finite width and closure of verbal subgroups in profinite groups.

Proposed method

• Uses the classification of finite simple groups and the solution to the Restricted Burnside Problem as foundational tools.
• Applies Proposition 1 from [NS] to bound the width of G^q in groups with bounded section involving Alt(k), using h(k,d,q) as the width bound.
• Employs Proposition 2 to control the structure of perfect normal subgroups N with large simple quotients, showing that G_q^*m generates N modulo a small subgroup.
• Constructs a characteristic series K1 ≥ K3 ≥ K4 ≥ K5 in G = H^q with K5 acceptable, enabling application of the Key Theorem (Proposition 4) to bound K5 in terms of products of commutators and qth powers.
• Combines bounds from Proposition 1, Proposition 2, and the Key Theorem to derive a uniform bound f(d,q) on the width of H^q in terms of d and q.
• Uses Schreier’s formula and the bound β(d,q) on the order of the d-generator restricted Burnside group to control the number of generators of H^q.

Experimental results

Research questions

• RQ1 Can the width of the verbal subgroup G^q be uniformly bounded in terms of the number of generators d and the exponent q for finite groups?
• RQ2 Is the verbal subgroup G^q open in any finitely generated profinite group, and does this follow from finite width of G^q in finite quotients?
• RQ3 Does the property that G^q has finite width in finite d-generator groups imply that G^q is closed in the profinite completion?
• RQ4 Can the result be extended to non-commutator words w, showing that w(G) is open in finitely generated profinite groups?

Key findings

• For any d, q ∈ ℕ, there exists f(d,q) such that every element of G^q in a d-generator finite group G is a product of at most f(d,q) qth powers.
• The verbal subgroup G^q is closed in any finitely generated profinite group G, and hence open, as a consequence of finite width in finite quotients.
• The result implies that the word x^q is d-restricted for every d, meaning that w(G) has finite width and is closed in any d-generator profinite group.
• The bound f(d,q) is constructed via a combination of the Restricted Burnside Problem, the classification of finite simple groups, and structural decomposition of G^q into acceptable subgroups.
• The number of qth powers needed to generate G^q is bounded by δ(d,q) = d·β(d,q)·f(d,q), where β(d,q) is the order of the d-generator restricted Burnside group of exponent q.
• The proof relies on the Key Theorem (Proposition 4) which expresses an acceptable normal subgroup K5 as a product of commutators and qth powers, with exponents depending only on d and q.