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the symmetric group s sub n is a group of permutations on a set with n elements a permutation is just a rearrangement of the set in this notation the s stands for symmetric and the n tells you the size of the set being permuted there are n factorial ways to permute a set with n elements so s sub n is a finite group with n factorial elements lets see some examples s3 is a group of permutations on a set with three elements while you can use any set to keep things simple well use the integers one two and three there are three factorial ways to permute this set these six permutations are the elements of the group s3 but what is the operation how do you combine two permutations consider the permutation two three one this permutation takes one two three and replaces it with two three one one is replaced with two two is replaced with 3 and 3 is replaced with 1. when viewed this way we see that a permutation acts like a function more specifically it is a bijection from the set 1 to 3 to its