Oliver subgroup

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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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Definition

The main Oliver subgroup

Suppose p is a prime number and P is a finite p-group. The Oliver subgroup of P, denoted X(P) (don't know what that letter actually is) is defined as the unique largest subgroup of P such that there exists an ascending series of subgroups of X(P):

 \{ e \} = Q_0 \le Q_1 \le \dots \le Q_n = X(P)

such that:

  1. Each Q_i is a normal subgroup of P.
  2. We have the condition that for all i

\! [\Omega_1(C_P(Q_{i-1})),Q_i;p-1] = \{ e \}

where the ;p - 1 indicates an iterated commutator of the form [[[ \dots [\Omega_1(C_P(Q_{i-1})),Q_i],Q_i],\dots,Q_i] with Q_i occurring p - 1 times.

Here, \Omega_1 denotes the first omega subgroup.

Other Oliver subgroups

Suppose p is a prime number and P is a finite p-group. The k^{th} Oliver subgroup of P, denoted X_k(P) (don't know what that letter actually is) is defined as the unique largest subgroup of P such that there exists an ascending series of subgroups of X(P):

 \{ e \} = Q_0 \le Q_1 \le \dots \le Q_n = X(P)

such that:

  1. Each Q_i is a normal subgroup of P.
  2. We have the condition that for all i

\! [\Omega_1(C_P(Q_{i-1})),Q_i;k-1] = \{ e \}

where the ;k - 1 indicates an iterated commutator of the form [[[ \dots [\Omega_1(C_P(Q_{i-1})),Q_i],Q_i],\dots,Q_i] with Q_i occurring k - 1 times.

Here, \Omega_1 denotes the first omega subgroup.

Relation between definitions

The main Oliver subgroup is the p^{th} Oliver subgroup.