On structure of an augmentation quotient group with N_p-series

Theorem 1 in this paper is taken from the paper by G. Losey and N. Losey⁵. We consider Theorem 1 to be highly significant and apply it to obtaining the rank of a certain augmentation quotient group by proposing Theorem 3 and giving its complete proof. We now state Theorem 3 as follows: 'Let G be nonabelian elementatry finite p-group (p ′ p≠2 with order p⁴, and let H be N_p-series of G with t₁ = 3, t₂ = 1, c = 2. Then, Q_n(G) is an abelian elementary p-group with rank 1/2(p+1)(p^Z+p+1) for all n≥3p-2'.