§ Normal subgroups through the lens of actions

• finite group is permutation subgroup
• ghg' is relavelling by g
• if gHg' = H, then H does not care about labelling
• thus H treats everyone uniformly
• prove that if H is normal, then if s in fix(H) then orb(s) in fix(H)
• when is stab(S) normal?when Stab(gx) equals g Stab(x) g' ?

• topology onS: closed sets are the common fixpoints of a set of group elements.