Solvable Groups in which Every Real Element has Prime Power Order

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 O₂(G)>1 and the case O₂(G)=1. Specifically we proved that if O₂(G)>1 then G is a \2,p\-group. Finally, by taking into consideration the examples presented in the analysis of the O₂(G)=1 case, we deduce some interesting and unexpected results about the connectedness of the real prime graph ΓR(G). In particular, we found that there are groups such that ΓR(G) has respectively 3 and 4 connected components.