On some infinite groups
A group G with a finite set of generators X and a set of defining relations \(R_ i=1\), \(i\in I\), can be represented by means of graphs using van Kampen diagrams. Following these ideas \textit{A. Yu Ol'shanskij} found examples of groups with many extraordinary properties [Geometry of defining relations in groups (Moscow, Nauka, 1989; Zbl 0676.20014)]. The author gives for sufficiently large odd integers \(n>2\cdot 10^{77}\) a series of examples of infinite simple groups G of exponent n with almost arbitrary lattices of subgroups. In particular, there exists an infinite simple group G of exponent n with descending chain condition for subgroups which does not satisfy the weak ascending chain condition for subgroups. There exists an infinite simple group G of exponent n which satisfies the ascending chain condition for subgroups and does not satisfy the descending chain condition. Similar examples exist in the class of torsion-free groups.
scientific article; zbMATH DE number 4200478