On a generalization of the Hughes subgroup
By definition, \(G\in{\mathcal H}_p(n)\) whenever \(G\) has \(n\) proper subgroups whose set-theoretic union is not \(G\) and all elements outside the union are of order \(p\). The author extends his previous results [Algebra Logika 19, 118-129 (1980; Zbl 0475.20018), Mat. Sb., Nov. Ser. 130(172), No. 1(5), 120-127 (1986; Zbl 0608.20025), and Sib. Mat. Zh. 30, No. 6(178), 208-218 (1989; Zbl 0727.20015)] on groups in \({\mathcal H}_p(1)\) to the case of \({\mathcal H}_p(n)\), \(n\leq p\). Typical results are as follows: (1) Suppose that \(G\) is a soluble \(p\)-group of derived length \(s\) in \({\mathcal H}_p(n)\) with \(n\leq p\). Then \(G\) has a normal subgroup of index dividing \(p^n\) which is nilpotent of class at most \((p^s-1)/(p-1)\). (2) Suppose that \(G\) is a soluble \(p\)-group of derived length \(s\) of exponent \(p^m\) in \({\mathcal H}_p(n)\) with \(n\leq p\). Then \(G\) is nilpotent of class bounded in terms of \(p\), \(s\), and \(m\).
scientific article; zbMATH DE number 1431689