[Paper Review] On solubility of groups with finitely many centralizers

Mohammad Zarrin|arXiv (Cornell University)|May 3, 2012

Finite Group Theory Research3 citations

TL;DR

This paper establishes that all finite groups with at most 21 centralizers are soluble, proving a sharp bound at n = 21. It further shows that if |G| < (30n + 15)/19, then G is non-nilpotent soluble, providing a partial solution to a conjecture by Ashrafi on the structure of finite Cₙ-groups with small order relative to n.

ABSTRACT

For any group G, let C(G) denote the set of centralizers of G. We say that a group G has n centralizers (G is a Cn-group) if |C(G)| = n. In this note, we prove that every finite Cn-group with n ? 21 is soluble and this estimate is sharp. Moreover, we prove that every finite Cn-group with |G| &lt; 30n+15 19 is non-nilpotent soluble. This result gives a partial answer to a conjecture raised by A. Ashrafi in 2000.

Motivation & Objective

• To determine the solubility of finite groups with finitely many centralizers, particularly for small values of n.
• To establish order bounds on Cₙ-groups that imply non-nilpotent solubility.
• To provide a partial answer to Ashrafi’s 2000 conjecture on the structure of finite Cₙ-groups with |G| ≤ 3n/2.
• To explore the relationship between the number of centralizers and group properties such as nilpotency and solubility.

Proposed method

• Uses the concept of centralizers C_G(g) for g ∈ G, defining Cₙ-groups as those with exactly n distinct centralizers.
• Applies Proposition 2.1 to show that any Cₙ-group satisfies the (𝒜, n−1) condition, meaning no set of n elements can all pairwise generate non-abelian subgroups.
• Employs Theorem 2.2 to bound |G : Z(G)| ≤ c^{n−1} for some constant c, linking group center size to n.
• Applies Lemma 2.3 to relate |G| and |I(G)| (involutions) via the inequality n ≤ (|G| + |I(G)|)/2.
• Uses Potter’s 1988 theorem on automorphisms inverting more than 4|G|/15 elements to deduce solubility.
• Analyzes group actions on centralizer sets to prove finiteness and order bounds for semi-simple Cₙ-groups.

Experimental results

Research questions

• RQ1 Is every finite Cₙ-group soluble when n ≤ 21, and is this bound sharp?
• RQ2 Can a bound on |G| in terms of n guarantee that a Cₙ-group is non-nilpotent and soluble?
• RQ3 Does the condition |G| < (30n + 15)/19 imply non-nilpotent solubility for finite Cₙ-groups?
• RQ4 To what extent does the number of centralizers constrain the structure of finite groups, especially regarding nilpotency and solubility?
• RQ5 How does the size of the set of involutions I(G) relate to the number of centralizers n in a finite group?

Key findings

• Every finite Cₙ-group with n ≤ 21 is soluble, and this bound is sharp, as A₅ has 22 centralizers and is simple.
• The alternating group A₅ is the smallest non-soluble Cₙ-group, confirming that n = 22 is the threshold for solubility.
• If |G| < (30n + 15)/19, then G is non-nilpotent and soluble, providing a quantitative solubility criterion.
• For any Cₙ-group G, the index |G : Z(G)| is bounded above by c^{n−1} for some constant c > 0.
• The set of involutions I(G) satisfies |I(G)| > (4|G|/15) − 1 whenever |G| < (30n + 15)/19, enabling application of Potter’s theorem.
• Semi-simple Cₙ-groups are finite and satisfy |G| ≤ (n−1)!, derived from group action on centralizer sets and nilpotency constraints.