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