On some problems regarding $LCM$-groups

Let $G$ be a finite group and denote by $o(g)$ the order of an element $g\in G$. We say that $G$ is an $LCM$-group if $o(x^ny)$ is a divisor of the least common multiple of $o(x^n)$ and $o(y)$ for all $x, y\in G$ and $n\in\mathbb{N}$. This paper extends some existing results on $LCM$-groups, such as the structure of a minimal non-$LCM$-group, and establishes criteria for $G$ to be an $LCM$-group or a nilpotent group. We also prove that, in general, a minimum cover of a finite set of $LCM$-groups is not an $LCM$-group, and we answer two questions posed by M. Amiri.

💡 Research Summary

The paper investigates finite groups through a divisibility condition on element orders, introducing the notion of an LCM‑group: a group G satisfies o(xⁿy) | lcm(o(xⁿ), o(y)) for every x, y∈G and every natural number n. This condition links naturally to previously studied CP₂‑groups (where o(xy)≤max{o(x),o(y)}) and P*₂‑groups (where the omega subgroups Ωₖ(G) consist exactly of the elements of order at most pᵏ). Lemma 2.2 shows that a group is an LCM‑group precisely when it is nilpotent and each Sylow subgroup is a CP₂‑group; for p‑groups this is equivalent to being a CP₂‑group.

The first major contribution is a structural description of minimal non‑LCM‑groups. Theorem 2.4 proves that any minimal non‑LCM‑group is either a minimal non‑LCM p‑group or a Schmidt group G = P⋊Q where the normal Sylow p‑subgroup P itself is an LCM‑group. This extends earlier results that required additional hypotheses on the Fitting subgroup. The paper also raises the open problem of characterising minimal non‑LCM p‑groups (Question 2.5).

Theorem 2.6 provides four equivalent characterisations for p‑groups: (i) being an LCM‑group, (ii) being a CP₂‑group, (iii) Ωₖ(G) = { x | x^{pᵏ}=1 } for all k, and (iv) being a P₂‑group. Consequently, the classes of minimal non‑LCM p‑groups, minimal non‑CP₂ p‑groups and minimal non‑P₂ p‑groups coincide. Proposition 2.7 shows that a non‑P₂ p‑group whose proper sections are all P₂‑groups is automatically a minimal non‑LCM p‑group, while the converse fails (e.g., Q₁₆).

To quantify how far a group is from being an LCM‑group, the authors introduce the ratio lcm(G)=|LCM(G)|/|G|, which equals 1 exactly for LCM‑groups. Theorem 3.6 demonstrates that no universal constant c∈(0,1) can serve as a threshold: for any c there exist both even‑order and odd‑order non‑LCM groups with lcm(G)>c, constructed via families Gₖ=D₈×C_{2ᵏ} and Lₖ=G×C_{3ᵏ}. However, by considering the minimum of lcm over all sections, lcm* (G)=min{lcm(S) | S a section of G}, one can obtain useful sufficient conditions.

The paper also answers two questions posed by M. Amiri. The first asks whether certain patterns in the order sequence force nilpotency; using Lemma 3.1–3.4 and the counterexamples above, the authors show the answer is negative. The second asks whether bounds on ψ(G)=∑_{x∈G}o(x) guarantee the LCM property; again, minimal non‑LCM groups provide counterexamples, showing that ψ alone cannot detect the property.

Throughout, the authors provide explicit calculations (often via GAP) of the size of LC M(G) for specific families, enumerate minimal non‑LCM p‑groups for small primes and exponents, and discuss the behavior of the newly defined ratios. The work bridges the study of element‑order based invariants with classical group‑theoretic structures (nilpotency, Schmidt groups, Frobenius groups) and introduces quantitative tools that may prove valuable for future investigations into the interplay between element orders and group structure.