Lower exponent-p central series

Definition

Suppose $p$ is a prime number and $G$ is a finite p-group. The lower exponent-p central series, also called the p-central series, of $G$ is a series ${\lambda }_n(G)$, $n\in \mathbb{N}$, defined as follows:

• ${\lambda }_1(G)=G$
• ${\lambda }_{n+1}(G)=[G,{\lambda }_n(G)]{\mho }^1({\lambda }_n(G))$

Here, ${\mho }^1({\lambda }_n(G))=({\lambda }_n(G){)}^p$ is the subgroup generated by the $p^{th}$ powers of the elements from ${\lambda }_n(G)$.

It is the fastest descending exponent-p central series.

Relation with other series

Corresponding ascending series

For a finite p-group, the corresponding ascending series, the upper exponent-p central series, is the socle series.

Other related series

The following series are closely related:

• Frattini series
• Jennings series

Subgroup series properties

Property	Meaning	Satisfied?	Proof
fully invariant series	all the member subgroups are fully invariant subgroups	Yes	lower exponent-p central series is fully invariant
strongly central series	descending series $G_m$ where $[G_m,G_n]\le G_{m+n}$ for all $m,n$	Yes	lower exponent-p central series is strongly central