On representation of infinite Groups as Automorphism Groups of Graphs

Author: Archil Kipiani
Keywords: Group, automorphisms group of graph, isomotphizm
Annotation:

The subject of discussion is the representation of infinite groups as automorphism groups of graphs. This question originates from D.Konig’s problem (see [1]). Using some results of the papers [2] and [3] we prove the following theorems. Theorem 1. Let l be any infinite cardinal and G is a group such as |G|£l , then there exists a family of pairwise non isomorphic connected graphs {H i :iÎI} such that, for each iÎI we have the following: • Aut ( H i ) ≅ G. • | H i | = l; • | I | =2 l; Theorem 2. There exists an infinite group, which is not isomorphic with none of the automorphisms group of a partially mono-unary algebra. References [1] D. König, Theotie der endlichen und unendlichen Graphen. Kombinatorische Topologie der Streckenkomplexse, Mathematik in Monographien 16. Akademische Verlagsgesellschaft, Leipzig, 1936. [2] A. Kipiani, On one uniform subset in a× aBulletin of Academy of Sciences of the Georgia,no. 2 (1989) (in Russian). [3] A. Kipiani, On automorphism groups of -Trees, Georgian Mathematical Journal , T. 15, No. 1, 2008.

Lecture files:

Archil Kipiani anot_eng(1).dok [en]
Archil Kipiani anot_geo(2).dok [ka]