Lower exponent-p central series

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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:

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