Conjugation family

Definition
Let $$G$$ be a finite group and $$P$$ be a $$p$$-Sylow subgroup of $$G$$, for some prime $$p$$. A conjugation family for $$S$$ in $$G$$ is a family $$\mathcal{F}$$ of subgroups of $$P$$, such that the following holds.

In terms of right actions
Whenever $$A$$ and $$B$$ are subsets of $$P$$ such that $$A^g = B$$ for some $$g \in G$$:

Then, $$A$$ and $$B$$ are $$\mathcal{F}$$-conjugate via $$g$$. In other words, we can find elements $$g_1,g_2,\ldots,g_n \in G$$, and subgroups $$T_1,T_2,\ldots,T_n \in \mathcal{F}$$ such that:


 * The subgroup generated by $$A$$ is in $$T_1$$
 * Each $$g_i$$ is in the normalizer of $$T_i$$ i.e. $$g_i \in N_G(T_i)$$
 * $$g = g_1g_2 \dots g_n$$
 * We have for any $$1 \le r < n$$:

$$\langle A \rangle^{g_1g_2 \dots g_r} \subseteq T_{r+1}$$

In terms of left actions
Whenever $$A$$ and $$B$$ are subsets of $$P$$ such that $$gAg^{-1} = B$$ for some $$g \in G$$:

Then, $$A$$ and $$B$$ are $$\mathcal{F}$$-conjugate via $$g$$. In other words, we can find elements $$g_1,g_2,\ldots,g_n \in G$$, and subgroups $$T_1,T_2,\ldots,T_n \in \mathcal{F}$$ such that:


 * The subgroup generated by $$A$$ is in $$T_1$$
 * Each $$g_i$$ is in the normalizer of $$T_i$$ i.e. $$g_i \in N_G(T_i)$$
 * $$g = g_ng_{n-1}\ldots g_1$$
 * We have for any $$1 \le r < n$$:

$$g_rg_{r-1} \ldots g_1 \langle A \rangle g_1^{-1}g_2^{-1} \ldots g_r^{-1} \subseteq T_{r+1}$$