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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 52 Issue: 1

Actions and semi-direct products in categories of groups with action

Tamar Datuashvili(Mathematical Institute of I. Javakhishvili Tbilisi State University)0000-0002-9956-7773
Tunçar Şahan(AKSARAY UNIVERSITY)0000-0002-6552-4695
Sayfa
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Yayın Tarihi
2023-02-15
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Yapay Zeka Hızlı Bakış & Çıkarımlar

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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

The results obtained in this paper will be applied in the forthcoming paper on the representability of actions in the category $\mathbf{rGr}^{\bullet}$.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Derived actions in the category of groups with action on itself $\mathbf{Gr}^{\bullet}$ are defined and described.
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Temel Bulgular & Veri:This category plays a crucial role in the solution of two problems of Loday stated in the literature.
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Sonuç & Katkı:The results obtained in this paper will be applied in the forthcoming paper on the representability of actions in the category $\mathbf{rGr}^{\bullet}$.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Group with action · semi-direct product · action

Abstract

Derived actions in the category of groups with action on itself $\mathbf{Gr}^{\bullet}$ are defined and described. This category plays a crucial role in the solution of two problems of Loday stated in the literature. A full subcategory of reduced groups with action $\mathbf{rGr}^{\bullet}$ of $\mathbf{Gr}^{\bullet}$ is introduced, which is not a category of interest but has some properties, which can be applied in the investigation of action representability in this category; these properties are similar to those, which were used in the construction of universal strict general actors in the category of interest. Semi-direct product constructions are given in $\mathbf{Gr}^{\bullet}$ and $\mathbf{rGr}^{\bullet}$ and it is proved that an action is a derived action in $\mathbf{Gr}^{\bullet}$ (resp. $\mathbf{rGr}^{\bullet}$) if and only if the corresponding semi-direct product is an object of $\mathbf{Gr}^{\bullet}$ (resp. $\mathbf{rGr}^{\bullet}$). The results obtained in this paper will be applied in the forthcoming paper on the representability of actions in the category $\mathbf{rGr}^{\bullet}$.

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Bu Makaleye Atıf Yap

Datuashvili, T., Şahan, T. (2023). Actions and semi-direct products in categories of groups with action. Hacettepe Journal of Mathematics and Statistics, Volume: 52 Issue: 1. https://doi.org/10.15672/hujms.1028848

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