Solvable groups in which every real element has prime power order

Abstract We study the finite solvable groups 𝐺 in which every real element has prime power order. We divide our examination into two parts: the case O 2 ⁢ (G) > 1 O₂ (G) >1 and the case O 2 ⁢ (G) = 1 O₂ (G) =1. Specifically we prove that if O 2 ⁢ (G) > 1 O₂ (G) >1, then 𝐺 is a 2, p \2, p\ -group. Finally, by taking into consideration the examples presented in the analysis of the O 2 ⁢ (G) = 1 O₂ (G) =1 case, we deduce some interesting and unexpected results about the connectedness of the real prime graph Γ R ⁢ (G) ₑ (G). In particular, we find that there are groups such that Γ R ⁢ (G) ₑ (G) has 3 or 4 connected components.