# Socle series

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

## Definition

Suppose is a group. The **socle series** of is an ascending series of subgroups , , defined as follows:

- is the trivial subgroup of .
- For , is the unique subgroup of such that is the socle of .

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

- For any limit ordinal , is the union of , .
- For the successor ordinal to , is the socle of .

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 .

### For a finite p-group

For a finite p-group, and more generally, for a nilpotent p-group, coincides with (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.