Oliver subgroup

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.