A progression-free group subset is a subset containing no nontrivial sequence , where belong to the group and nontrivial means , with the identity element (Solymosi 2013, Sabatini 2026). In an additively written Abelian group, this is the arithmetic progression . The first and last terms may coincide when has order 2. Thus a group in which every element other than the identity element has order 2 admits no such progression-free subset of more than one element.

For a finite group , let be the largest cardinality of a progression-free subset. Sabatini (2026) proved the asymptotic upper bound

as , for absolute constants . The result applies to arbitrary finite groups, including noncommutative ones.

See also

Arithmetic Progression, Finite Group, Nonaveraging Sequence

Explore with Wolfram|Alpha

References

Sabatini, L. "Progression-Free Sets in Class and Arbitrary Finite Groups." 10 Nov 2026. https://arxiv.org/abs/2611.14464v1.Solymosi, J. "Roth-Type Theorems in Finite Groups." Europ. J. Combin. 34, 1454-1458, 2013. https://doi.org/10.1016/j.ejc.2013.05.027.

Cite this as:

Weisstein, Eric W. "Progression-Free Group Subset." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Progression-FreeGroupSubset.html

Subject classifications