We study subgroups G of the multiplicative group of a unital Dedekind σ-complete partially ordered linear algebra A satisfying the property that the set H = {x 2 : x ∈ G} is a chain. Among other things, we show that if such a group is bounded above by some element u ∈ A, then x 4 = 1 for all x ∈ G (so that H consists entirely of involutions), while if the upper bound u belongs to G, then x 2 = 1 for all x ∈ G (so that G is abelian).