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This is a Quantitative Study study.

October 5, 2025Open Access

Solvable Groups in which Every Real Element has Prime Power Order

Key Points

If O2(G) > 1, G is a {2,p}-group, highlighting a specific structural requirement.
Interesting results about the real prime graph demonstrate unexpected connectedness in group components.
The study divides the problem into cases for O2(G) over and under 1, revealing distinct group properties.
Identified groups indicate that the real prime graph can have varying numbers of connected components.

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 ₑ (G). In particular, we found that there are groups such that ₑ (G) has respectively 3 and 4 connected components.

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Cite This Study

Alessandro De Giorgi (Wed,) studied this question.

synapsesocial.com/papers/68e24e59d6d66a53c2472e98 https://doi.org/https://doi.org/10.48550/arxiv.2504.07327

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