Conjugation family

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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}