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

Authors : Tamar DATUASHVİLİ, Tunçar ŞAHAN

Pages : 103-113

Doi:10.15672/hujms.1028848

View : 20 | Download : 6

Publication Date : 2023-02-15

Article Type : Research Paper

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 $\\mathbf{Gr}^{\\bullet}$ insert ignore into journalissuearticles values(resp. $\\mathbf{rGr}^{\\bullet}$); if and only if the corresponding semi-direct product is an object of $\\mathbf{Gr}^{\\bullet}$ insert ignore into journalissuearticles values(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