A permutation group on a set $A$ is a subgroup of the symmetric group $S_A$ .

fact: conjugating stabilizer of an element by permutation produces stabilizer of permuted element

1:

$A$ set

2:

$a\in A$

3:

$X$ permutation group on $A$

4:

$\sigma\in X$

5:

$\sigma \mathrm{Stab}_X(a) \sigma^{-1}= \mathrm{Stab}_X(\sigma(a))$

fact: if a permutation group acts transitively, then the intersection of conjugated stabilizers is the identity

1:

$A$ set

2:

$a\in A$

3:

$X$ permutation group on $A$

4:

$\cap_{\sigma\in X} \sigma \mathrm{Stab}_X(a)=1$

Note: This is a ``seed'' entry written using a short-hand format described in this FAQ.