On the Structure of the Augmentation Quotient Group for Some Non-Abelian $p$-Groups

Abstract

In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian $p$-group which has a cyclic subgroup of index $p$, where $p$ is an odd prime, and $k$ is greater than or equal to 3. A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.