- #1

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I'm on page 31 now (of this book: http://www.math.miami.edu/~ec/book/).

The question is: Show that if there is a bijection between X and Y , there is an isomorphism between S(X) and S(Y).

I see the bijection btw S(X) and S(Y) in my head, but can't find the right mathematical symbols to express it. The bijection would take a permutation in S(X) and assign to it the permutation in S(Y) that is ""identical in essence"" to it. I.e. if we somehow manage to order the sets X and Y, then the bijection from S(X) to S(Y) assigns to the permutation in S(X) that warps the n^th element of X to the m^th element of X, the i^th element to the j^th element, and so on, to the permutation of S(Y) that warps the n^th element of Y to the m^th element of Y, the i^th element to the j^th element, and so on.

The question is: Show that if there is a bijection between X and Y , there is an isomorphism between S(X) and S(Y).

I see the bijection btw S(X) and S(Y) in my head, but can't find the right mathematical symbols to express it. The bijection would take a permutation in S(X) and assign to it the permutation in S(Y) that is ""identical in essence"" to it. I.e. if we somehow manage to order the sets X and Y, then the bijection from S(X) to S(Y) assigns to the permutation in S(X) that warps the n^th element of X to the m^th element of X, the i^th element to the j^th element, and so on, to the permutation of S(Y) that warps the n^th element of Y to the m^th element of Y, the i^th element to the j^th element, and so on.

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