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. $g_{i}\in N_{G}(T_{i})$
• $g=g_{1}g_{2}\dots g_{n}$
• We have for any $1\leq r<n$:

$\langle A\rangle ^{g_{1}g_{2}\dots g_{r}}\subseteq T_{r+1}$

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. $g_{i}\in N_{G}(T_{i})$
• $g=g_{n}g_{n-1}\ldots g_{1}$
• We have for any $1\leq r<n$:

$g_{r}g_{r-1}\ldots g_{1}\langle A\rangle g_{1}^{-1}g_{2}^{-1}\ldots g_{r}^{-1}\subseteq T_{r+1}$