A(t)-GROUPS SATISFYING A CHAIN CONDITION

作者：Zhang, Lihua; Qu, Haipeng^*

来源：Journal of Algebra and Its Applications, 2014, 13(4): 1350137.

DOI：10.1142/S0219498813501375

• At-groups
• metacyclic p-groups
• minimal non-metacyclic p-groups
• chain condition

For a positive integer t, a finite p-group G is an At-group if all subgroups of index p(t) in G are abelian, and at least one subgroup of index p(t-1) in G is not abelian. An A(t)-group G satisfies a chain condition if every A(i)-subgroup of G is contained in an A(i+1)-subgroup for all i is an element of {0, 1, 2,..., t - 1}, where A(0)-subgroups denote abelian subgroups. We prove that if t >= 2, then, except for some p-groups of small order, the following conditions for a group G of order p(n) are equivalent: (1) G is an A(t)-group satisfying a chain condition; (2) every subgroup of order p(n-k) in G is an A(t-k)-subgroup for 0 <= k <= t; (3) G is an ordinary metacyclic group.

• 出版日期2014-6
• 单位北京邮电大学; 山西师范大学