Socle series

Definition
Suppose $$G$$ is a group. The socle series of $$G$$ is an ascending series of subgroups $$\operatorname{Soc}^i(G)$$, $$i \in \mathbb{N}$$, defined as follows:


 * $$\operatorname{Soc}^0(G)$$ is the trivial subgroup of $$G$$.
 * For $$i > 0$$, $$\operatorname{Soc}^i(G)$$ is the unique subgroup of $$G$$ such that $$\operatorname{Soc}^i(G)/\operatorname{Soc}^{i-1}(G)$$ is the socle of $$G/\operatorname{Soc}^{i-1}(G)$$.

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


 * For any limit ordinal $$\alpha$$, $$\operatorname{Soc}^\alpha(G)$$ is the union of $$\operatorname{Soc}^\beta(G)$$, $$\beta < \alpha$$.
 * For $$\beta + 1$$ the successor ordinal to $$\beta$$, $$\operatorname{Soc}^{\beta + 1}(G)/\operatorname{Soc}^\beta(G)$$ is the socle of $$G/\operatorname{Soc}^\beta(G)$$.

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, $$\operatorname{Soc}(G)$$ coincides with $$\Omega_1(Z(G))$$ (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.