Socle series

This article defines a quotient-iterated series with respect to the following subgroup-defining function: socle

Definition

Suppose $G$ is a group. The socle series of $G$ is an ascending series of subgroups $\stackrel{\mathrm{o}}{\mathrm{S}}(G)$, $i\in \mathbb{N}$, defined as follows:

• $\stackrel{\mathrm{o}}{\mathrm{S}}(G)$ is the trivial subgroup of $G$.
• For $i>0$, $\stackrel{\mathrm{o}}{\mathrm{S}}(G)$ is the unique subgroup of $G$ such that $\stackrel{\mathrm{o}}{\mathrm{S}}(G)/\stackrel{\mathrm{o}}{\mathrm{S}}(G)$ is the socle of $G/\stackrel{\mathrm{o}}{\mathrm{S}}(G)$.

The series can be extended to a transfinite series. In that case, the definition is as follows for ordinals:

• For any limit ordinal $\alpha$, $\stackrel{\mathrm{o}}{\mathrm{S}}(G)$ is the union of $\stackrel{\mathrm{o}}{\mathrm{S}}(G)$, $\beta <\alpha$.
• For $\beta +1$ the successor ordinal to $\beta$, $\stackrel{\mathrm{o}}{\mathrm{S}}(G)/\stackrel{\mathrm{o}}{\mathrm{S}}(G)$ is the socle of $G/\stackrel{\mathrm{o}}{\mathrm{S}}(G)$.

Since socle is strictly characteristic, the socle series is a strictly characteristic series, i.e., all members of the series are strictly characteristic subgroups of $G$.

For a finite p-group

For a finite p-group, and more generally, for a nilpotent p-group, $\mathrm{S}\mathrm{o}\mathrm{c}(G)$ coincides with ${\mathrm{\Omega }}_1(Z(G))$ (see socle equals Omega-1 of center in nilpotent p-group). The socle series in this case is also called the upper exponent-p central series, and is the fastest ascending exponent-p central series. The corresponding fastest descending exponent-p central series is the lower exponent-p central series.