Finiteness of the cyclic group related to the group inverse of a matrix and nite Markov chains Cenap ZEL & Hanifa ZEKRAOUI Faculty of Sciences, University of Abant Izzet Baysal, Bolu Turkey. cenap.ozel@gmail.com Faculty of Exacts and natural Sciences, University of Larbi-Ben-Mhidi, Oum-El-Bouaghi, Algeria. hanifazekraoui@yahoo.fr March 29, 2013 Abstract The group inverse is one of the generalized inverses possessing the properties the closest to the normal inverse. The positive and negative powers of a given matrix A (the latter being interpreted as powers of A # ; the group inverse of A), together with the projection AA # as the unit element, form an Abelian group. In this paper, we give some equivalent conditions so that the group is nite, and apply this result to nite Markov chains. Key words: Generalized inverse, Group inverse, Finite cyclic group; Markov chains; Matrix index; Stochastic matrix. AMS subject classication: 15A09, 15A03, 15A23 . 1 Introduction. The most of the properties of the generalized inversion were handled in the work of A. Ben Israºl and T. N. E. Greville [1], also in the work of Z. Nashed [3]. Some algebraic structures on the set of generalized inverses of matrices are studied in [4]. Some algebraic properties are widely studied in [5]. The group inverse is one of the generalized inverses possessing the properties the closest to the usual inverse. The positive and the negative powers of a given matrix A (the latter being interpreted as powers of A # the group inverse of A), together with the projection AA # as the unit element, form an Abelian group. The rst main result of this paper, is giving some equivalent conditions so that this group is nite and the second one is applying this result to nite Markov chains. 1