# Oliver subgroup

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.