On a non-solvable group satisfying xG=(x−1)G

Abstract

A group G satisfies Syskin’s condition if elements of same order are conjugates. If a group G satisfies Syskin’s condition, then each element and its inverse are conjugate to each other, i.e., for all x∈G, x^G=(x⁻¹)^G, but not conversely. Thus, the class of those groups satisfying Syskin’s condition forms a proper subclass of groups satisfying x^G=(x⁻¹)^G. In this note, it is proved that if a group G meets the condition x^G=(x⁻¹)^G, then G cannot be of odd order. As the main result, it is shown that if x^G=(x⁻¹)^G holds for a centreless and non-solvable group G of order 120 such that G≠G’, then G≌S5.