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 ${\mathbf{\text{O}}}_2(G)>1$ and the case ${\mathbf{\text{O}}}_2(G)=1$. Specifically we proved that if ${\mathbf{\text{O}}}_2(G)>1$ then $G$ is a $\{2,p\}$-group. Finally, by taking into consideration the examples presented in the analysis of the ${\mathbf{\text{O}}}_2(G)=1$ case, we deduce some interesting and unexpected results about the connectedness of the real prime graph ${\mathrm{\Gamma }}_{\mathbb{R}}(G)$. In particular, we found that there are groups such that ${\mathrm{\Gamma }}_{\mathbb{R}}(G)$ has respectively 3 and 4 connected components.