A classification of 2-generator \(p\)-groups \(p\geq 3\) with many subgroups 2-subnormal
In this paper, using results of C. Hobby, we show that if \(G\) is a 2-generator \(p\)-group, \(p\geq 3\), in which every cyclic subgroup is 2-subnormal, then in G the normalizer of every subgroup is a normal subgroup and \(G\) is a homomorphic image of \(G(p,r)\), where (i) If \(p>3\), then \(G(p,r)=<x,y| [x,y,x]=x^{p^ r}\), \([x,y,y]=y^{p^ r}\), \(x^{p^{2r}}=y^{p^{2r}}=1\), \(G_ 4=1>\), \(r>1;\) (ii) \(G(3,1)=<x,y| [x,y,x]=x^ 3=[x,y,y]=y^ 3\), \(y^ 9=1\), \(G_ 4=1>;\) (iii) If \(r>1\), then \(G(3,r)=<x,y| [x,y,x]=x^{3^ r}=[x,y,y]=y^{3^ r}\), \(y^{3^{2r}}=x^{3^{2r-1}}=1\), \(G_ 4=1>\).
scientific article; zbMATH DE number 3847662