Search results for "Locally finite group"
showing 2 items of 52 documents
Finite groups with all minimal subgroups solitary
2016
We give a complete classification of the finite groups with a unique subgroup of order p for each prime p dividing its order. All the groups considered in this paper will be finite. One of the most fruitful lines in the research in abstract group theory during the last years has been the study of groups in which the members of a certain family of subgroups satisfy a certain subgroup embedding property. The family of the subgroups of prime order (also called minimal subgroups) has attracted the interest of many mathematicians. For example, a well-known result of Itˆo (see [8, Kapitel III, Satz 5.3; 9]) states that a group of odd order with all minimal subgroups in the center is nilpotent. Th…
On the Frattini subgroup of a finite group
2016
We study the class of finite groups $G$ satisfying $\Phi (G/N)= \Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.