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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. The '''permutation model''' consists of all hereditarily symmetric elements, and is a model of ZFA.
==Construction of filters on a group==
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 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''.
==References==
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