Positivity Results for Cyclic Subgroups

• Cyclic subgroups are subgroups generated by a single element that reveal key structural and quantitative properties across various algebraic systems.
• Enumerative positivity results establish sharp lower bounds and uniqueness criteria, illustrating that non-cyclic groups exceed the cyclic minimal counts.
• Positivity phenomena extend to representation theory, elliptic curves, and combinatorial structures, underpinning cyclic subgroup separability and group decompositions.

A cyclic subgroup is a subgroup generated by a single element. Positivity results for cyclic subgroups encompass both enumerative inequalities and structural properties asserting the abundance, prevalence, or stability of cyclic subgroups within algebraic or combinatorial contexts. These results arise across finite group theory, representation theory, algebraic geometry, geometric group theory, and algebraic combinatorics, often providing sharp characterizations and effective lower bounds.

1. Enumerative Positivity: Counting Cyclic Subgroups in Finite Groups

For a finite group $G$ of order $n$, denote by $c_m(G)$ the number of cyclic subgroups of order $m$, and let $c(G) = \sum_{m|n} c_m(G)$ be the total number of cyclic subgroups. In the landmark result of (Garonzi et al., 2015), it is shown that among all groups of order $n$, the cyclic group $C_n$ uniquely minimizes the number of cyclic subgroups: $c(G) \geq c(C_n) = d(n),$ where $d(n)$ is the number of positive divisors of $n$. Equality holds if and only if $G$ is cyclic. The proof leverages Frobenius's theorem on solution counts of $x^m=1$ (ensuring each solution count is divisible by $m$) and applies Möbius inversion, which, through positivity-weighted sums, forces all non-cyclic structures to have strictly more cyclic subgroups.

The same work also considers the invariant

$\alpha(G) = \frac{|C(G)|}{|G|},$

where $C(G)$ is the set of cyclic subgroups, establishing that for any finite group,

$\alpha(G) \leq \alpha(Z(G)),$

with equality characterized precisely via structural constraints involving $G/Z(G)$ and the orders of elements in the cosets, leading to ramifications for the structure being essentially 2-central or 4-abelian for equality (Tărnăuceanu, 2020).

2. Extremality and Cyclicity Detection via Element-Order Products and Generalized Sums

Another positivity result regards the extremal behavior of the product of element orders: $P(G) = \prod_{x \in G} o(x),$ where $o(x)$ is the order of $x$. It is proved that

$P(G) \leq P(C_n),$

with equality if and only if $G$ is cyclic (Garonzi et al., 2015). The proof uses divisor sums and explicit combinatorial weights to show the extremal property of the cyclic group.

Beyond these, a general two-parameter family of invariants is introduced: $$R_G(r,s) = \sum_{x \in G,\, o(x)|n} \frac{o(x)^s}{\varphi(o(x))^r},$$ where $\varphi$ is Euler’s totient function. For $r=s<0$, nilpotent groups maximize $R_G(r,r)$, with cyclic groups being the unique maximizers in certain parameter regimes. These inequalities not only detect cyclicity and nilpotency, but, by parameter choice, test other structure, raising further open questions on broader group properties detectable through such positivity frameworks.

3. Positivity of Branching Coefficients for Cyclic Subgroups in Representations

Let $C_n \leq S_n$ be the cyclic subgroup generated by an $n$-cycle, and let $\delta^r$ denote its linear characters. For the irreducible $S_n$-characters $\chi_\lambda$, the branching coefficient for the $C_n$-isotypic component is

$$a^r_\lambda = \langle \mathrm{Res}^{S_n}_{C_n} \chi_\lambda, \delta^r \rangle_{C_n}.$$

The main result of (S, 16 Dec 2025) is a complete classification of positivity: except for finitely many explicit partitions $\lambda$ (the “extremal” shapes: $(n)$, $(1^n)$, $(n-1,1)$, $(2,1^{n-2})$ and, for $n$ even, two more shapes at $r=0,n/2$), all coefficients $a^r_\lambda$ satisfy

$a^r_\lambda > \frac{n}{6}$

for all $n \geq 11$ and for all $r$. Furthermore, $a^r_\lambda = 0$ only for these extremal cases and $a^r_\lambda = 1$ only on the “first hooks”. This result both unifies and strengthens earlier lower bounds, being valid uniformly for all remaining partitions and characters. The proof combines combinatorial methods (Littlewood–Richardson structures, partition reduction, and modular branching) and specializes broader branching to the cyclic subgroup.

Partition $\lambda$	$a^r_\lambda = 0$ for some $r$?	$a^r_\lambda = 1$ for some $r$?
$(n),\ (1^n)$	Yes	No
$(n-1,1),\ (2,1^{n-2})$	No	Yes
Others	No	No

4. Positivity for Galois-Stable Cyclic Subgroups on Elliptic Curves

Let $N_i(X)$ denote the number of $\mathbb{Q}$-isomorphism classes of elliptic curves of height at most $X$ with at least $i$ pairs of Galois-stable cyclic subgroups of order $4$. It is proved in (Pomerance et al., 2020) that the counts exhibit positive main terms: $$N_1(X) = c_{1,1}X^{1/3} + c_{1,2}X^{1/6} + O(X^{0.105}), \quad N_2(X) = c_{2,1}X^{1/6} + o(X^{1/12}),$$ with explicitly computable constants $c_{1,1}>0,\ c_{1,2}<0,\ c_{2,1}>0$. For $i>2$, $N_i(X)=0$ since algebraically, only up to two pairs can be Galois-stable over $\mathbb{Q}$. The positivity of the main terms establishes not only non-trivial existence but density results for such elliptic curves.

Techniques include explicit parametrization via binary forms, geometry-of-numbers estimates leveraging Huxley's strong Lipschitz principle, and careful analytic control of error terms through Möbius inversion and Dirichlet series, all ensuring the precise positivity of the asymptotic growth.

5. Positivity in Geometric Group Theory: Separability and Stability

A group $G$ is cyclic subgroup separable (CSS) if every cyclic subgroup is closed in the profinite topology; for a prime $p$, $G$ is $p$-CSS if every $p$-isolated cyclic subgroup is closed in the pro-$p$ topology. In (Berlai et al., 2018), it is shown that the CSS property is preserved under graph products: if all vertex groups in a finite simplicial graph product are CSS, the graph product is again CSS. Similarly, $p$-CSS is preserved under graph products within the class $\Upsilon$ of torsion-free groups with unique roots and primitive stability. Every right-angled Artin group is thus CSS and $p$-CSS for every $p$.

Classification of $p$-isolated cyclic subgroups in right-angled Artin groups is explicit: a cyclic subgroup is $p$-isolated if and only if it is generated by a primitive element raised to a $p^e$-th power. The result gives a positivity property for closure/separability: the abundance and stability of cyclic subgroups under group-theoretic constructions.

6. Positivity of Cyclic Decompositions: Active Sums and Metacyclic Groups

The notion of the active sum of a family $\mathcal{F}$ of subgroups generalizes the direct sum for abelian groups. For $G$ a finite group, the active sum $S$ of a discrete family $\mathcal{F}$ of cyclic subgroups is constructed as the quotient of the free product modulo relations encoding conjugacy and action, with a canonical surjection $S \to G$. For this surjection to be an isomorphism, $\mathcal{F}$ must be regular and independent.

In (Díaz-Barriga et al., 2014), it is shown that every finite metacyclic group can be reconstructed as the active sum of a discrete family of cyclic subgroups, positively realizing $G$ as built entirely from cyclic subgroups. Concretely, both split and non-split metacyclic $p$-groups, and groups with Hall decomposition, can always be realized this way. This provides a comprehensive positivity result within the framework of group decompositions, contrasting with known negative results for certain simple groups.

7. Positivity Phenomena in Algebraic and Combinatorial Representations

Positivity for cyclic subgroups also arises in enumerative and geometric combinatorics. In (Adin et al., 2022), new cyclic descent extensions for involutions, standard Young tableaux, and Motzkin paths are constructed, with explicit actions by the cyclic group. The associated quasisymmetric generating functions display Schur-positivity: they expand as symmetric functions with nonnegative coefficients in the Schur basis, an intrinsically positive phenomenon aligning with deep representation-theoretic patterns.

Through explicit bijections and cyclic group actions, the construction ensures that positivity features—such as equidistribution and symmetry—emerge for these statistics, reinforcing the structural role of cyclic subgroups in algebraic combinatorics.

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In summary, positivity results for cyclic subgroups—whether in enumeration, representation branching, algebraic stability, or geometric constructions—provide sharp tools for characterizing group and module structure, detecting extremality, guaranteeing explicit lower bounds, and constructing group-theoretic and combinatorial objects with desirable properties. These results interconnect diverse areas including finite group theory, representation theory, algebraic geometry, geometric group theory, and algebraic combinatorics, often providing the essential foundation for deeper structural insights.