Lower exponent-p central series ends in {1} for finite p-groups

Question

For a finite $p$-group $G$ the lower exponent $p$-central series $(\lambda_i(G))_i$ is inductively defined as

• $\lambda_1(G) = G$,
• $\lambda_{i+1}(G) = [G,\lambda_i(G)](\lambda_i(G))^p$.

Since finite $p$-groups are nilpotent the lower central series $(G_i)_i$

• $G_1=G$,
• $G_{i+1}=[G,G_i]$

terminates in {1}. In general $\lambda_2(G) = [G,G]G^p \not\subset [G,G] =G_2$ and inductively $\lambda_n(G) \not\subset G_n$ (note $G_n \subset \lambda_n(G)$). However, in what I've been able to find on the lower exponent $p$-central series, it seems like this series also ends in {1} for all finite $p$-groups. Is this true? And does anyone know how to see this?

Thank you for your help

Answer 1

Yes it's true. Any finite $p$-group has a central series, and this can be refined to a central series in which all quotients are elementary abelian (you could even have all quotients of order $p$). Since the lower exponent $p$-central series is the fastest descending such central series, it must lie below the one we have just constructed, and hence terminate in $\{ 1 \}$.