Ein Untergruppensatz für modulare Gruppen
Recently A. Yu. Ol'shanskiĭproved the existence of so-called ''Tarski-monsters'', i.e. infinite groups with every proper subgroup of prime order [Math. USSR, Izv. 16, 279-289 (1981), translation from Izv. Akad. Nauk SSR, Ser. Mat. 44, 309--321 (1980; Zbl 0475.20025)]. A group \(G\) is called modular if the lattice of subgroups, \(L(G)\), of \(G\) is modular. In the paper under review we prove that every Infinite periodic 2-generator modular group \(G\) with generators \(a\) and \(b\), such that the cyclic subgroups generated by \(a\) and \(b\) intersect trivially, contains a Tarski-monster as a subgroup. An immediate consequence of the above is the fact that such a group is the direct sum of finitely many Tarski-monsters and a finite modular group; furthermore the orders of elements in different direct summands are relatively prime. Thus by a well-known theorem of Suzuki \(L(G)\) has a corresponding decomposition.
infinite groups with every proper subgroup of prime order
scientific article; zbMATH DE number 3758578