Communications in Algebra· 2026Q2
On some problems regarding LCM-groups
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- Q2SCImago
- 2026year
Short summary
A finite group G is an LCM-group if the order of xny divides the LCM of o(xn) and o(y) for all elements x, y and natural number n. This paper establishes criteria for groups to be LCM-groups or nilpotent, and shows that a minimum cover of LCM-groups is generally not an LCM-group.
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Key points
- Defines LCM-groups based on the order of elements xny in relation to o(xn) and o(y).
- Establishes criteria for determining if a finite group is an LCM-group or a nilpotent group.
- Extends known results concerning minimal non-LCM-groups.
- Proves that a minimum cover of LCM-groups is generally not an LCM-group.
AI-generated from the title and abstract; the full text is not read.
Abstract
Let G be a finite group and denote by o(g) the order of an element g∈G. We say that G is an LCM-group if o(xny) is a divisor of the least common multiple of o(xn) and o(y) for all x,y∈G and n∈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.
The authors' abstract, as published at the source. Communications in Algebra, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics