If S is a subset of a group G, we define its square S^2 by the formula S^2 = {ab | a, b ∈S}. We prove that if S is a finite subset of an ordered group that generates a nonabelian group, then\ud the order of S^2 is bigger or equal to 3|S|-2. This generalizes a classical result from the theory of set addition. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INDAM

Denote the sum of element orders in a finite group G by ψ(G) and let C n denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that ψ(G) ≤ 7 11 ψ(C n ) and ψ(G) < 1 q−1 ψ(C n ). The first result is best possible, since for each n = 4k, k odd, there exists a group G of order n satisfying ψ(G) = 7 11 ψ(C n ) and the second result implies that if G is of odd order, then ψ(G) < 1 2 ψ(C n ). Our results improve the inequality ψ(G) < ψ(C n ) obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some ψ(G)-based sufficient conditions for the solvability of G. the elements of G, there are many new and old results as well as many open 5 questions concerning ω(G) (see for example [9]). In [1] H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs introduced the function ψ(G), which denotes the sum of element orders of a finite group G, and proved that if G is a non-cyclic group of order n then ψ(G) < ψ(C n ), where C n denote the cyclic group of order n. Recently S.M. Jafarian Amiri and M. Amiri in [5] (see also [2] and [4]) and R.

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Abstract.We look at groups which have no (nonabelian) free subsemigroups. It is known that a finitely generated solvable group with no free subsemigroup is nilpotent-by-finite. Conversely nilpotent-by-finite groups have no free subsemigroups. Torsion-free residually finite-p groups with no free subsemigroups can have very complicated structure, but with some extra condition on the subsemigroups of such a group one obtains satisfactory results. These results are applied to ordered groups, right-ordered groups, and locally indicable groups.