This attractive paper considers for a periodic group \(G\) its subgroup \(\operatorname{LC}G := \langle \operatorname{LCM}(G) \rangle\) where \(\operatorname{LCM}(G)\) is its subset \(\{ x\in G \mid o(x^nz)\) divides \(\operatorname{lcm} (o(x^n), o(z))\) for all \(z\in G\) and all \(n\in \mathbb{N} \}\) with \(o\) being the order function on \(G\)’s elements and \(\operatorname{lcm}\) denoting the function of the least common multiple of its arguments. If \(\operatorname{LCM}(G) = G\) then \(G\) is called an \(\operatorname{LCM}\)-group and it is later inquired after all finite groups \(G\) with \(\operatorname{LC}G = \operatorname{LCM}(G)\). The authors adopt the notation established in the splendid book on finite group theory by I. M. Isaacs [Finite group theory. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1169.20001)].
The paper also considers the exciting class \(\operatorname{CP}\) of (locally) finite groups all of whose elements have prime power order (equivalently, where the centraliser of every nontrivial element is a \(p\)-group for some prime \(p\) which depends on the element). Finite \(\operatorname{CP}\)-groups were introduced by G. Higman [J. Lond. Math. Soc. 32, 335–342 (1957; Zbl 0079.03204)] and the locally finite \(\operatorname{CP}\)-groups were classified by A. L. Delgado and Y.-F. Wu [Ill. J. Math. 46, No. 3, 885–891 (2002; Zbl 1021.20029)]. Delgado’s and Wu’s paper presents an overview of the fascinating history of the classification of finite \(\operatorname{CP}\)-groups.
The paper has three sections: “1. Introduction”, “2. \(\operatorname{LCM}(G)\)” and “3. \(LC\)-series”. It was published on 28 October 2021 on arXiv [arXiv:2009.01044] with these sections and the additional section “4. \(\operatorname{LCM}\)-graph”.
We cannot help to first criticise a few “misprints”: Theorem 1.1. and Theorem 2.10. talk about “locally finite periodic” groups; but a locally finite group is obviously periodic since its cyclic subgroups are finite; therefore these theorems should talk just about “locally finite” groups; the last sentence in Definition 2.2. (where the definition of \(\operatorname{LCM}(G)\), given in the introduction, is missing) and the proof of Corollary 2.24. should not be written in italics; in the title the second “the” should be deleted: “The order of the product of two elements in periodic groups”.
The authors recall the subset \(\operatorname{CP}2\) of \(\operatorname{CP}\) which was introduced and thoroughly studied by M. Tărnăuceanu [Publ. Math. Debr. 80, No. 3–4, 457–463 (2012; Zbl 1261.20028)]: \(\operatorname{CP}2\) is the class of finite groups \(G\) such that \(o(xy)\leq \max \{ o(x), o(y) \}\) for all \(x, y\in G\). They state Tărnăuceanu’s heady characterisation of \(\operatorname{CP}2\) as Theorem 2.1. and that they shall need it but it remains unclear where they use it. As their main result, the authors prove that \(\operatorname{LC}G\) is in a locally finite group \(G\) a locally nilpotent characteristic subgroup of \(G\) and base their proof on the corresponding quite profound theorem for finite groups. They then give a complicated example of an infinite group \(G\) where \(\operatorname{LC}G\) is not (locally) nilpotent which is based on work by B. Bekka et al. [Kazhdan’s property. Cambridge: Cambridge University Press (2008; Zbl 1146.22009)], M. Gromov [Publ., Math. Sci. Res. Inst. 8, 75–263 (1987; Zbl 0634.20015)] and A. Yu. Ol’shanskij [Int. J. Algebra Comput. 3, No. 4, 365–409 (1993; Zbl 0830.20053)]. Previously they prove the considerably interesting theorem that a finite group is an \(\operatorname{LCM}\)-group if and only if it is nilpotent and all its Sylow subgroups are \(\operatorname{CP}2\)-groups.
Finally, they study LC-series of a group \(G\) defined by \(\operatorname{LC}G =: \operatorname{LC}_1(G)\leq \operatorname{LC}_2(G)\leq \dots \leq \operatorname{LC}_i(G)\leq \dots\) where \(\operatorname{LC}_i(G)/\operatorname{LC}_{i -1}(G) := \operatorname{LC}(G/LC_{i -1}(G))\) for \(i = 2, 3, \dots\). They say that \(G\) is an LC-nilpotent group if it has a finite LC-series all of whose factors are nilpotent and state three examples. If \(G\) is a finite LC-nilpotent group, \(H\) is the direct product of its Sylow subgroups and \(\operatorname{LC}(G/\operatorname{LC}_{i -1}(G)) = \operatorname{LCM}(G/\operatorname{LC}_{i -1}(G))\) for all \(i\), they show that there exists a bijection \(f\) from \(G\) to \(H\) such that \(o(x)\) divides \(o(f (x))\) for all \(x\in G\). They close with stating two corollaries and an example and with posing two questions, namely asking for all finite groups \(G\) with \(\operatorname{LC}G = \operatorname{LCM}(G)\) and for all LC-nilpotent groups of class two.