Question about Group Theory - Symmetric Set of Permutations

Modern Algebra
Group Theory (XI)
Symmetric Set of Permutations

In S3 give an example of two elements x,y such that (x.y)^2 is not equal to x^2.y^2.

The fully formatted problem is in the attached file.

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Solution Summary

Symmetric sets of permutations are investigated. The solution is detailed and well presented.