Socle series

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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 \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.