A note on a class of generalized nilpotent groups introduced by Bechtell and Doerk. Issue 10 (2nd October 2020)

• Record Type: Journal Article
• Authors: Kirtland, Joseph
• Is Part Of: Communications in algebra. Volume 48:Issue 10(2020)
• Page Start: 4142
• Page End: 4148
• Publication Date: 2020-10-02
• Subjects: Frattini subgroup -- nilpotent group -- Sylow p-subgroups; PRIMARY 20F19 -- Secondary: 20D20; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1756310
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: All groups are finite with Φ ( G ) denoting the Frattini subgroup of a group G . If G is nilpotent with subgroups H and K where H ≤ K, then Φ ( H ) ≤ Φ ( G ) and Φ ( H ) ≤ Φ ( K ) . However, as demonstrated by the symmetric group S 3, there are non-nilpotent groups that also satisfy these two properties. In 1965 H. Bechtell introduced a class of groups that satisfy the property that Φ ( H ) ≤ Φ ( G ) for all subgroups H of a group G . About 30 years later Doerk introduced a class of solvable groups that satisfy the property Φ ( H ) ≤ Φ ( K ) when H ≤ K ≤ G for a group G . These two classes are identical when restricted to solvable groups. In this short paper, we will extend the work done by Bechtell and Doerk by presenting some additional properties and structural results concern this class of solvable groups.