November 9, 2025Open Access

Profinite Direct Sums with Applications to Profinite Groups of Type ΦR

Key Points

Profinite groups exhibit closure under subgroup formation for type Φ_R, and this highlights their intricate algebraic structure.
The primary focus lies on demonstrating properties of profinite direct sums within the context of profinite modules, including projectiveness.
Exploration of profinite direct sums guides an understanding of subgroup characteristics of profinite groups of type Φ_R, enriching modular theory.
This analysis supports the idea of finding broader applications of profinite structures in abstract algebra and group theory.

Abstract

We show that the "profinite direct sum" is a good notion of infinite direct sums for profinite modules having properties similar to direct sums of abstract modules. For example, the profinite direct sum of projective modules is projective, and there is a Mackey's Formula for profinite modules described using these sums. As an application, we prove that the class of profinite groups of type ΦR is closed under subgroups.

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Cite This Study

J. Tang (Mon,) studied this question.

synapsesocial.com/papers/690fdcdaf60c54d04ea37f77 https://doi.org/https://doi.org/10.48550/arxiv.2504.05182