Finite extensions of Abelian groups with minimum condition

Abstract

Among the groups with minimum condition one meets, often quite unexpectedly, groups which satisfy one of the following two conditions: (a) the commutator subgroup is finite; (b) there exists an abelian subgroup of finite index. It is our objective in this investigation to give various characterizations of these two classes of groups some of which have very little obvious connection with either the minimum condition or properties (a) and (b). Our principal result, stated in ?0, contains characterizations of the groups with minimum condition and finite commutator subgroup; and the proof of this theorem is effected in ?1 to ?5. In ?6 we specialize this result to show that torsion groups with finite automorphism groups are finite. In ?7 we enunciate a characterization of the groups with minimum condition possessing abelian subgroups of finite index; and the proof of this proposition will be effected in ?8 to ?11. These results are used in ?12 to show that the p-Sylow subgroups of these groups are conjugate. 0. In this section we enunciate and discuss our principal