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.