Solvable Groups in which Every Real Element has Prime Power Order

2025/04/09 by Giorgi, Alessandro
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2504.07327

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 O₂(G)>1 and the case O₂(G)=1. Specifically we proved that if O₂(G)>1 then G is a \2,p\-group. Finally, by taking into consideration the examples presented in the analysis of the O₂(G)=1 case, we deduce some interesting and unexpected results about the connectedness of the real prime graph Γ_ℝ(G). In particular, we found that there are groups such that Γ_ℝ(G) has respectively 3 and 4 connected components.

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