Baniasad Azad & Khosravi Supersolubility Criterion

• The paper introduces a supersolubility criterion based on the normalized sum of element orders, establishing that groups with ψ'(G) > 31/77 are supersoluble.
• The methodology employs structural, arithmetic, and combinatorial techniques, including Frobenius automorphism actions and fixed-point centralizers, to analyze group properties.
• The work provides a comprehensive classification of finite groups, distinguishing between supersoluble families and marginal cases through explicit bounds and subgroup analyses.

The supersolubility criterion of Baniasad Azad and Khosravi gives an explicit group-theoretic condition for the supersolubility of finite groups in terms of the normalized sum of element orders, as well as insights into the role of Frobenius automorphism groups and fixed-point centralizers. The criterion has been fully classified and connected to structural group properties in recent work, particularly via the bound $\psi'(G) > 31/77$, where $\psi'(G)$ denotes the normalized sum of element orders for finite group $G$ (Iorio et al., 16 Jan 2026). The methods and conclusions also appear in the context of Frobenius automorphism actions (Tang et al., 2015), further enriching understanding of group solubility and associated invariants.

1. Definition of the Criterion

Let $G$ be a finite group, and define

$\psi(G) = \sum_{x \in G} o(x)$

as the sum of the orders of all elements in $G$, with $o(x)$ the order of $x \in G$. The normalized sum is

$\psi'(G) = \frac{\psi(G)}{\psi(\mathcal{C}_{|G|})}$

where $\mathcal{C}_n$ is the cyclic group of order $n$.

The Baniasad Azad–Khosravi criterion states:

If

$\psi'(G) > \psi'(A_4) = \frac{31}{77}$

then $G$ is supersoluble (Iorio et al., 16 Jan 2026). Equivalently, $A_4$ is the minimal non-supersoluble group, and any group $G$ exceeding the normalized sum of $A_4$ must be supersoluble.

For groups admitting a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that $C_G(F)=1$, the supersolubility of $G$ follows if the centralizer $C_G(H)$ is supersoluble and $C_{G'}(H)$ (centralizer within the derived subgroup) is nilpotent (Tang et al., 2015).

2. Theoretical Background and Key Lemmas

The proof framework relies on structural, arithmetic, and combinatorial ingredients:

• Multiplicativity: For $A$ and $B$ of coprime order, $\psi'(A \times B) = \psi'(A) \psi'(B)$.
• Cyclic group formula: For $n = \prod p_i^{\alpha_i}$,

$$\psi(\mathcal{C}_n) = \prod_{i} \frac{p_i^{2\alpha_i+1}+1}{p_i+1}$$

• Semidirect product formula: If $P \trianglelefteq G$ is a cyclic Sylow subgroup and $G = P \rtimes H$,

$\psi(G) = |P|\psi(H) + (\psi(P)-|P|)\psi(C_H(P))$

• CyclBoundIndex: If $\psi'(G) > r/s$, there exists $x$ such that

$$|G : \langle x \rangle| < \frac{s}{r} \prod_{p \mid |G|} \frac{p+1}{p}$$

• Core-lemma: For cyclic $A < G$, $|A:K| < |G:A|$ with core $K$.

In the context of Frobenius group actions, essential lemmas include:

• If $C_G(F) = 1$ under nilpotent $F$, then $G$ is soluble.
• For $FH$-invariant normal $N \trianglelefteq G$, $C_{G/N}(H) = C_G(H) N/N$.
• The unique $FH$-invariant Hall $T$-subgroups form a system.
• Transfer of solubility properties from $C_G(H)$ and $C_{G'}(H)$ to $G$ under the specified fixed-point structure.

3. Classification of Groups Satisfying the Criterion

Groups $G$ with $\psi'(G) > 31/77$ are precisely classified by their structure as follows (Iorio et al., 16 Jan 2026):

Family	Representative Groups	Condition
“Large” M-groups	$\mathcal{C}_n$, $Q_8$, various semidirect products & their coprime cyclic extensions	Always supersoluble
Extremal non-M-groups	$D_8 \times \mathcal{C}_m$, $$(\mathcal{C}_7 \rtimes_\iota \mathcal{C}_2) \times \mathcal{C}_m$$	$\psi'(G) = 19/43 > 31/77$
Near-equality cases	$Q_{16}$, $\mathcal{C}_3 \times \mathcal{C}_3$, etc.	$31/77 < \psi'(G) < 19/43$ (not supersoluble)

All supersoluble groups fall in the first category, while the extremal cases provide boundaries for the modular subgroup lattice property. Groups with normalized sums strictly between $31/77$ and $19/43$ are not supersoluble.

4. Methodology of Proof and Structural Implications

The argument proceeds via induction and reduction:

• Lower bounds on $\psi(\mathcal{C}_n)$ provide constraints on possible quotient indices.
• Application of Lucchini’s core-lemma yields normal cyclic subgroups.
• If $\psi'(G) > 31/77$, $G$ possesses a normal cyclic Sylow 2- or $p$-subgroup $(p = \max \pi(G))$.
• Decomposition $G = P \rtimes H$ allows comparison of $\psi(G)$ and $\psi(H)$, employing the semidirect formula.
• Case analysis on possible $H$ (with inductive bounds) restricts the possibilities, leading to the explicit family structure in the previous section.

For Frobenius automorphism groups, the stepwise transfer of solubility and supersolubility properties through centralizers enables the general result: if $C_G(H)$ is supersoluble and $C_{G'}(H)$ is nilpotent, then $G$ is supersoluble (Tang et al., 2015).

5. Equality and Sharpness

The bound $\psi'(G) > 31/77$ is optimal. The only groups with $\psi'(G) = 31/77$ are $A_4$ and its direct products with coprime cyclics. These mark the exact threshold at which supersolubility fails.

Groups lying just above $31/77$ but below $19/43$ are rare and precisely enumerated; they exhibit structures that are not supersoluble but approach the modular lattice property—$Q_{16}$, $\mathcal{C}_3 \times \mathcal{C}_3$, $D_{12}$, etc.

6. Connections to Hall Systems and Permutation Criteria

The role of normalized sum bounds is intimately connected to Hall subgroup systems—groups with NS-supplemented Sylow subgroups, where the existence of such supplements implies permuting Hall subgroups, whence $p$-solubility and supersolubility by induction (Monakhov et al., 2019). The existence of a Frobenius group of automorphisms, as in (Tang et al., 2015), introduces additional control through centralizer properties and the transfer of structural characteristics from fixed-point subgroups.

A plausible implication is that normalized element order criteria, Frobenius automorphism actions, and NS-supplementation instantiate parallel approaches to characterizing supersolubility through diverse but converging invariants and subgroup permutation properties.

7. Concluding Remarks and Extensions

The supersolubility criterion of Baniasad Azad and Khosravi provides an explicit spectral bound for group supersolubility, with complete classification for groups exceeding the critical normalized sum. The framework allows for modular transfer between centralizer conditions in the presence of Frobenius automorphism groups and gives exact inclusion and exclusion criteria via explicit group families (Iorio et al., 16 Jan 2026, Tang et al., 2015). The combinatorial and permutation-based methodologies reflected in NS-supplementation and normalized sum bounds highlight a unified approach to finite group classification—in particular, those admitting strong solubility behaviors. This suggests the possibility of further refinements for related invariants and the extension to broader classes of linear or permutation groups.