Oliver 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 $X(P)$:

$\{e\}=Q_{0}\leq Q_{1}\leq \dots \leq Q_{n}=X(P)$

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 $X(P)$:

$\{e\}=Q_{0}\leq Q_{1}\leq \dots \leq Q_{n}=X(P)$

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.