RESTRICTED-FINITE GROUPS WITH SOME APPLICATIONS IN GROUP RINGS.

We carry out a study of groups G in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite restricted-finite abelian group is isomorphic to Z×K or Zp∞ ×K, where K is a finite group and p is a prime number. We also prove that a group G is infinitely generated restricted-finite if and only if G = AT where A and T are subgroups of G such that A is normal quasi-cyclic and T is finite. As an application of our results, we show that if G is not torsion with finite G′ and the group-ring RG has restricted minimum condition, then R is a semisimple ring and G ∼= T ⋊ Z, where T is finite whose order is unit in R. The converse is also true with certain conditions including G = T × Z. [ABSTRACT FROM AUTHOR]