Inverse of a Product of Group Elements (Socks-Shoes Property) | Abstract Algebra

We prove that the inverse of a product of group elements ab is equal to b^-1*a^-1. In other words, (ab)^-1 = b^-1*a^-1. This is sometimes called the Socks-Shoes property, because if we put on socks then shoes, then to take them off (the inverse) we have to reverse the order and take off shoes then socks. This is a straightforward proof directly from the definition of inverse element.

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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[talastra]

What would have been really helpful here would be a numerical example where ab doesn't look like multiplication.

[davidcontreras]

How exactly did you get (ab)(b^-1a^-1)? Like why did you just put (ab) on the left hand side?

[maxpercer]

4:00 why do you need to prove both directions, isnt it sufficient one