Solvable groups in which every real element has prime power order
Year: 2026
Type: article
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 O 2 ( G ) > 1 \mathbf{O}_{2}(G)>1 and the case O 2 ( G ) = 1 \mathbf{O}_{2}(G)=1 . Specifically we prove that if O 2 ( G ) > 1 \mathbf{O}_{2}(G)>1 , then 𝐺 is a { 2 , p } \{2,p\} -group. Finally, by taking into consideration the examples presented in the analysis of the O 2 ( G ) = 1 \mathbf{O}_{2}(G)=1 case, we deduce some interesting and unexpected results about the connectedness of the real prime graph Γ R ( G ) \Gamma_{\mathbb{R}}(G) . In particular, we find that there are groups such that Γ R ( G ) \Gamma_{\mathbb{R}}(G) has 3 or 4 connected components.
Source: Journal of Group Theory
Author: Alessandro Giorgi
Institution: University of Florence
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,712
OpenAlex ID: W7125514829