In mathematical set theory, a permutation model is a model of set theory with atoms (ZFA) constructed using a group of permutations of the atoms.
Permutation models were introduced by Fraenkel (1922) and developed further by Mostowski (1938).
A normal filter of G is a collection F of subgroups of G such that If V is a model of ZFA with A the set of atoms, then an element of V is called symmetric if the subgroup fixing it is in F, and is called hereditarily symmetric if it and all elements of its transitive closure are symmetric.
A filter on a group can be constructed from an invariant ideal on of the Boolean algebra of subsets of A containing all elements of A.
Here an ideal is a collection I of subsets of A closed under taking finite unions and subsets, and is called invariant if it is invariant under the action of the group G. For each element S of the ideal one can take the subgroup of G consisting of all elements fixing every element S. These subgroups generate a normal filter of G.