Abstract Algebra 15: The inverse of ab is b^{-1}a^{-1} (socks shoes property)

Abstract Algebra 15: The inverse of ab is b^{-1}a^{-1} (socks shoes property)

Abstract: We prove that the inverse of ab is b^{1}a^{-1}. This is sometimes called the "socks shoes" property since if you first put on your socks and then put on your shoes, then to undo that work you first take of your shoes before taking off your socks!

This video accompanies the class "Introduction to Abstract Algebra" at Colorado State University:

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Isn't it sufficient to show that b^{-1}a^{-1} is a left inverse of ab. Do we need to show that b^{-1}a^{-1} is also a right inverse of ab.