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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="bold">O</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> \mathbf{O}_{2}(G)>1 and the cas