Let be a set. Let be a collection of permutations of .

1. Show that there is a permutation group that contains .
2. Let be the intersection of all permutation groups that contain . Show that is a permutation group.
3. Why is the smallest permutation group that contains . In other words, if is any other permutation group that contains , why must we have .

• We could alternately define as the set of permutations of that we can express as a finite composition of elements in and inverses of elements in .

problem.permutation_group_generated_by_s.tex

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\begin{problem}
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• Generating set of a group