Two new functions related to the sum of element orders of a finite group

Marius T\u{a}rn\u{a}uceanu

Two new functions related to the sum of element orders of a finite group

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This note introduces two normalised sums of element orders attached to a normal subgroup and proves that one is bounded below and the other above by the corresponding quotient value, with equality fully classified.

desk verdict Correct, compact note introducing two relative invariants for sums of element orders; the main inequalities and equality cases hold, with an external classification that checks out. read the letter →

arxiv 2608.10035 v1 pith:VF6PNVSA submitted 2026-08-10 math.GR

Marius Tărnăuceanu This is my paper · ORCID

classification math.GR MSC 20D6020E34

keywords elementordersfinitegroupsequalorderpairsisolatedsubgroupssumofaveragecyclicp-groupsgeneralizedquaternion2-groups

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This note studies two numbers attached to a finite group $G$ and a proper normal subgroup $H$: the average order of the elements outside $H$, and a variant that divides the same sum by $|G|^2-|H|^2$. The main theorem proves that the first number cannot fall below the corresponding average in the quotient $G/H$, while the second cannot exceed its quotient analogue. It then characterizes exactly when equality occurs: for the first, precisely when $(G,H)$ is an equal order pair with $H$ an isolated subgroup; for the second, precisely when $G$ is a cyclic $p$-group with any proper nontrivial $H$, or a generalized quaternion $2$-group with $H$ its center. These inequalities yield two-sided bounds on $\psi(G)$ for group extensions, and the paper works out the case of ZM-groups as an example.

What carries the argument

The argument is carried by the coset decomposition $\psi(G)=\psi(H)+\sum_{i=2}^{m}\sum_{h\in H}o(x_i h)$, where $m=[G:H]$ and $x_iH$ runs through the non-trivial cosets of $H$, together with the choice of elements $y_i,z_i$ in each coset attaining the minimum and maximum of $o(x_i h)$. Sandwiching each inner sum between $o(y_i)|H|$ and $o(z_i)|H|$, then estimating $o(y_i)\ge o(y_iH)$ and $o(z_i)\le o(z_iH)|H|$, proves both inequalities directly. The equality analysis turns the collapsed sandwich into the pointwise conditions (3) and (4), identifies them as equal-order-pair conditions, and for the second inequality invokes Lemma 2.2: condition (4) forces $\langle x^n\rangle=H\subseteq\langle x\rangle$, so $H$ is comparable with every cyclic subgroup, and the cited classification restricts $G$ to cyclic $p$-groups and generalized quaternion $2$-groups.

What would settle it

Check all finite groups of order at most 64 by machine: for every proper nontrivial normal subgroup $H$, compare $\psi''(G,H)$ with $\psi''(G/H,1)$. Any equality outside the pairs $(C_{p^n}, C_{p^i})$ and $(Q_{2^n}, Z(Q_{2^n}))$ would refute the stated classification of equality cases; separately, a group with a nontrivial subgroup comparable with every cyclic subgroup but not cyclic or quaternion would refute the cited classification used in Lemma 2.2.

Watch

Extended reading notes

Core claim

The central claim is a pair of sharp inequalities for the normalised sums $o(G,H)=(\psi(G)-\psi(H))/(|G|-|H|)$ and $\psi''(G,H)=(\psi(G)-\psi(H))/(|G|^2-|H|^2)$, valid for every finite group $G$ and proper normal subgroup $H$: $o(G,H)\ge o(G/H,1)$ and $\psi''(G,H)\le \psi''(G/H,1)$. With $H$ non-trivial, equality in the first holds exactly when the elements of each coset $xH$ all have the same order as $x$ and $H$ meets every cyclic subgroup outside itself trivially, i.e. $(G,H)$ is an equal order pair with $H$ isolated. Equality in the second holds exactly when $G\cong C_{p^n}$ with $H$ an arbitrary proper nontrivial subgroup, or $G\cong Q_{2^n}$ with $H=Z(G)\cong C_2$.

Load-bearing premise

The load-bearing premise is the cited classification, not proved in this note, that a finite group with a nontrivial subgroup comparable with every cyclic subgroup under inclusion must be a cyclic $p$-group or a generalized quaternion $2$-group; if that theorem is wrong or its hypotheses are not met, the only-if direction of the equality characterization for $\psi''$ collapses.

Editorial extensions

If this is right

Under the hypotheses of Corollary 1.2, the theorem gives explicit bounds on $\psi(G)$: $\psi(H)+|H|(\psi(G/H)-1)\le \psi(G)\le \psi(H)+|H|^2(\psi(G/H)-1)$.
For $G=ZM(m,n,r)$ and $H=\langle a\rangle$, the bounds become $\psi(C_m)+m(\psi(C_n)-1)\le \psi(G)<\psi(C_m)+m^2(\psi(C_n)-1)$, with the lower bound an equality exactly under the arithmetic condition $d=n$ and $m_1\nmid r^{n_1}-1$ for all proper divisors $m_1,n_1$.
Equality in the first inequality characterizes isolated subgroups through a purely coset-order condition, giving a testable way to recognize them.
Equality in the second inequality isolates cyclic $p$-groups and generalized quaternion $2$-groups as the only extremal cases, i.e. the finite groups with a unique minimal subgroup.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the same coset-sandwich should apply to a chain of normal subgroups, giving telescoping products of relative averages and hence bounds on $\psi(G)$ for iterated extensions; the note treats only a single subgroup $H$.
Editorial extension: condition (4) could be studied without the equal-order-pair assumption; if equality in (2) already forces the two listed families under weaker hypotheses, that would show which part of Lemma 2.2 is really essential.
Editorial extension: the open problem on isolated-subgroup equal order pairs could be approached by tabulating normal subgroups $H$ of small groups such that $H$ is isolated; such a census would indicate how close the equality cases are to Frobenius groups.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a short, correct note in the sum-of-element-orders area. The new content is the pair of relative invariants o(G,H) and psi''(G,H), defined on a finite group with a proper normal subgroup, and Theorem 1.1 stating o(G,H) >= o(G/H,1) and psi''(G,H) <= psi''(G/H,1), with equality characterizations. The proof is a clean coset-wise min/max argument on the orders of elements in nontrivial cosets. I verified the steps: the inequalities reduce to comparing the min/max representatives to the quotient, and the equality cases follow by translating the equalities back to conditions on every element outside H. The first equality becomes 'equal order pair with H isolated'; the second becomes 'equal order pair satisfying o(x)=o(xH)|H|', and then Lemma 2.2 gives the dichotomy cyclic p-group / generalized quaternion 2-group. That lemma is the one non-self-contained step: it invokes Tarnauceanu's earlier characterization [19, Thm 1.1] of groups with a subgroup comparable to every cyclic subgroup. The dependency is load-bearing but the hypotheses are satisfied, and I found no misapplication. The classification itself is published peer-reviewed work, so this is a reasonable citation rather than circularity.

The corollaries for metabelian groups and ZM-groups follow directly. The ZM-group equality condition in Corollary 1.3 uses Lemma 2.1, whose proof is terse but valid: the equivalence between equal order pair and Camina pair there uses the known fact that cyclic subgroups in a ZM-group are conjugate exactly when they have equal orders, and the algebra checks out. The open problem at the end is natural.

Soft spots: this is a note, not a deep theory. The impact is limited to the specialized community working on psi(G) and related invariants. The reliance on [19] and [6] means a referee should double-check those classifications, but on inspection they fit. There are no data or code to evaluate. The paper self-cites the author's previous work heavily, but in context these citations are relevant and not circular.

The paper deserves a serious referee. I would accept it if the editor's scope includes short technical notes in group theory. It is not a major advance, but the result is correct, cleanly proved, and genuinely new.

Referee Report

0 major / 6 minor

Summary. The paper introduces two relative invariants for a finite group G and a proper normal subgroup H: o(G,H) = (ψ(G)-ψ(H))/(|G|-|H|) and ψ''(G,H) = (ψ(G)-ψ(H))/(|G|^2-|H|^2). The main result, Theorem 1.1, proves that o(G,H) ≥ o(G/H,1) and ψ''(G,H) ≤ ψ''(G/H,1), with equality characterized: equality in the first inequality holds exactly when (G,H) is an equal order pair with H an isolated subgroup, and equality in the second holds exactly when G is a cyclic p-group with H a proper nontrivial subgroup or G is a generalized quaternion 2-group with H its center. The proof of the inequalities uses a coset decomposition and min/max bounds; the equality cases reduce to lemmas, one of which relies on a published classification. Two corollaries apply the inequalities to metabelian groups and ZM-groups, and an open problem is posed about equal order pairs with isolated subgroups.

Significance. The central inequalities are correct and are proved by a short, self-contained min/max argument that is a natural extension of the standard sum-of-element-orders machinery. The equality characterizations are nontrivial and connect the new invariants to equal order pairs, isolated subgroups, and the known dichotomy of cyclic p-groups and generalized quaternion 2-groups. The paper is therefore a useful contribution to the literature on sums of element orders, even though the applications are modest and the main theorem is local in scope. The proof of the equality case in (2) depends on the author's earlier characterization [19]; I checked the hypotheses and the application appears sound for nontrivial H.

minor comments (6)

[Lemma 2.2] As stated, Lemma 2.2 is false when H=1: for every finite group G, (G,1) is an equal order pair satisfying (4), yet G need not be a cyclic p-group or a generalized quaternion 2-group. Please add the hypothesis that H is non-trivial to the statement (and to the proof, where the appeal to [19] requires a nontrivial breaking point).
[Lemma 2.1 and Corollary 1.3] These statements need the hypotheses m>1 and n>1. For m=1, ZM(1,n,r) is cyclic of order n and H=1; if n is a prime power, the claimed strict upper bound in Corollary 1.3 becomes an equality, and the equivalence in Lemma 2.1 fails because d=1 while 'G is a Frobenius group with kernel H' is false for a trivial kernel. The degenerate cases should be excluded or handled separately.
[Lemma 2.1, proof of a)⇒b)] The step with the notation '⟨b^x a^y⟩ = ⟨b^x⟩ α(u,v)' and the subsequent expression 'b^x a^y = b^{xz} a^{v(1-r^{xz})}' is very hard to follow and appears to contain a typo in the normal-form computation. Please rewrite this part with explicit normal forms and justify why z=1 follows.
[Theorem 1.1 statement] The notation 'H = p^{n-i}G' is nonstandard; it would be clearer to write that H is the unique subgroup of order p^i, or H = ⟨p^{n-i}⟩ in the cyclic group C_{p^n}.
[Corollary 1.2] The phrase 'Let G is a finite metabelian group' should be 'Let G be a finite metabelian group'.
[Introduction] The sentence about ψ''(G,H) providing 'finer structural control when investigating the boundary cases of Camina pairs and Frobenius groups' is vague; a concrete explanation of why the square normalization distinguishes the equality cases would help orient the reader.

Circularity Check

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No significant circularity: inequalities are proved directly from the coset decomposition, and the cited classification [19] is independent of the new invariants.

full rationale

The derivation of inequalities (1)-(2) is self-contained: equation (5) expresses psi(G)-psi(H) as a sum over nontrivial cosets of H, and the min/max bounds o(y_i)|H| <= sum_h o(x_i h) <= o(z_i)|H| together with o(y_iH) <= o(y_i) and o(z_i) <= o(z_iH)|H| give exactly the two inequalities; no fitted parameters, no assumption containing the target result, and no quotient property equivalent to the conclusion. The equality cases are also derived from the same coset decomposition: equality in (1) forces o(y)=o(yh) for all h in H and o(y)=o(yH), which by definition means (G,H) is an equal order pair with H isolated; equality in (2) similarly forces o(z)=o(zh) and o(z)=o(zH)|H|, i.e. condition (4). The only external input is Lemma 2.2's use of [19, Thm. 1.1] to classify groups in which H is comparable with every cyclic subgroup. That cited theorem is a published, parameter-free classification not involving the functions o(G,H) or psi''; it is independent support rather than a restatement of the present definitions or a fitted prediction. The self-citation is real but not circular. No step reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The central claim rests on standard group theory and published classification results from [6] and [19].

assumptions (5)

domain assumption Theorem A: For any finite group G of order n, psi(G) is at most psi(C_n) with equality iff G is cyclic.
Background result from [1] that frames the study; not used in the main proof.
domain assumption Theorem B: There is a bijection between G and C_n such that o(x) divides o(f(x)) for all x.
Cited from [3] as background; not invoked in the proof of Theorem 1.1.
domain assumption Camina's theorem: For a normal subgroup H with coprime |H| and [G:H], the conditions of a Camina pair and a Frobenius group coincide.
Used in Lemma 2.1 to show b) iff c) for ZM-groups.
domain assumption [19, Theorem 1.1]: A finite group in which a nontrivial subgroup is comparable with every cyclic subgroup under inclusion is either a cyclic p-group or a generalized quaternion 2-group.
Load-bearing for the equality characterization in Lemma 2.2; cited but not proved.
standard math Standard facts: for any element x in G and normal H, the order of xH divides the order of x, and o(x) is at most o(xH)|H|.
Used in the proof of Theorem 1.1.

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Pith. "Pith review of Two new functions related to the sum of element orders of a finite group." pith.science (2026). https://pith.science/paper/VF6PNVSA

@misc{pith2026260810035,
  author       = {Pith},
  title        = {Pith review of: Two new functions related to the sum of element orders of a finite group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VF6PNVSA}},
  note         = {Machine review of arXiv:2608.10035}
}

abstract

Let $G$ be a finite group and $H$ be a proper normal subgroup of $G$. In this note, we introduce two new functions $o(G,H)$ and $\psi''(G,H)$ involving the sums of element orders of $G$ and $H$. We prove that they satisfy certain inequalities, with equality if and only if $(G,H)$ is an equal order pair of a particular type.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [19]

T˘ arn˘ auceanu,A characterization of generalized quaternion2-groups, C.R

M. T˘ arn˘ auceanu,A characterization of generalized quaternion2-groups, C.R. Math., Acad. Sci. Paris348(2010), 731-733

work page 2010

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Camina,Some conditions that almost characterize Frobenius groups, Israel J

A.R. Camina,Some conditions that almost characterize Frobenius groups, Israel J. Math.31(1978), 153-160

work page 1978

[3] [1]

Amiri, S.M

H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs,Sums of element orders in finite groups, Comm. Algebra37(2009), 2978-2980

work page 2009

[4] [2]

Amiri and S.M

M. Amiri and S.M. Jafarian Amiri,Characterization of finite groups by a bijection with a divisible property on the element orders, Comm. Algebra 45(2017), 3396-3401. 9

work page 2017

[5] [3]

Amiri,On a bijection between a finite group and cyclic group, J

M. Amiri,On a bijection between a finite group and cyclic group, J. Pure Appl. Algebra228(2024), article ID 107632

2024

[6] [4]

Baniasad Asad and B

M. Baniasad Asad and B. Khosravi,A criterion for solvability of a finite group by the sum of element orders, J. Algebra516(2018), 115-124

work page 2018

[7] [5]

Baniasad Azad and B

M. Baniasad Azad and B. Khosravi,On two conjectures about the sum of element orders, Can. Math. Bull.65(2022), 30-38

work page 2022

[8] [7]

Camina, R.D

A.R. Camina, R.D. Camina, Mark L. Lewis, E. Pacifici, L. Sanus and M. Vergani,Group cosets with all elements of equal order, J. Algebra 688(2026), 694-717

work page 2026