Solvable Groups in which Every Real Element has Prime Power Order
Alessandro Giorgi
TL;DR
The paper investigates finite solvable groups G for which every real element has prime power order, by analyzing the real prime graph $\Gamma_{\mathbb{R}}(G)$. It develops a two-case framework based on $O_2(G)$: (i) when $O_2(G)>1$, G must be a $\{2,p\}$-group with a structured $2$-Frobenius decomposition, and (ii) when $O_2(G)=1$, the authors construct explicit examples showing that $\Gamma_{\mathbb{R}}(G)$ can have 3 or 4 connected components, challenging intuitive parallels with other real-graph variants. The work leverages quotient descent, DGN’s theorems (Theorem A, SH, (H2)) and Frobenius-type decompositions to constrain the group structure and to illustrate possible complexity beyond the $\{2,p\}$-case. These results illuminate how real-element constraints interact with solvable-group architecture and raise open questions about the maximal possible number of connected components of $\Gamma_{\mathbb{R}}(G)$. The paper also contrasts $\Gamma_{\mathbb{R}}(G)$ with related graphs $\Gamma_{cd,\mathbb{R}}(G)$ and $\Gamma_{cs,\mathbb{R}}(G)$, highlighting unique behaviors in the real-element setting.
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 $\textbf{O}_2(G)>1$ and the case $\textbf{O}_2(G)=1$. Specifically we proved that if $\textbf{O}_2(G)>1$ then $G$ is a $\{2,p\}$-group. Finally, by taking into consideration the examples presented in the analysis of the $\textbf{O}_2(G)=1$ case, we deduce some interesting and unexpected results about the connectedness of the real prime graph $Γ_{\mathbb{R}}(G)$. In particular, we found that there are groups such that $Γ_{\mathbb{R}}(G)$ has respectively 3 and 4 connected components.