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 <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> O₂(G)>1 and 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> O₂(G)=1 . Specifically we prove that if <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> O₂(G)>1 , then 𝐺 is a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{2,p\} -group. Finally, by taking into consideration the examples presented in the analysis of the <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> O₂(G)=1 case, we deduce some interesting and unexpected results about the connectedness of the real prime graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">Γ</m:mi> <m:mi mathvariant="double-struck">R</m:mi> </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:math> ΓR(G) . In particular, we find that there are groups such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">Γ</m:mi> <m:mi mathvariant="double-struck">R</m:mi> </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:math> ΓR(G) has 3 or 4 connected components.