- Hacettepe Journal of Mathematics and Statistics
- Vol: 52 Issue: 1
- Actions and semi-direct products in categories of groups with action
Actions and semi-direct products in categories of groups with action
Authors : Tamar Datuashvili, Tunçar Şahan
Pages : 103-113
Doi:10.15672/hujms.1028848
View : 12 | Download : 3
Publication Date : 2023-02-15
Article Type : Research
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 $\\mathbf{rGr}^{\\bullet}$.Keywords : Group with action, semi-direct product, action, extension